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Università di Tor Vergata – Dipartimento di Ingegneria Civile ed Ingegneria Informatica
Exercises in Mathematical Analysis I
Alberto Berretti, Fabio Ciolli
2
1 Fundamentals
1.1 Polynomial inequalities
Solve the following inequalities for x ∈ R:
Ex. 1. (x3− 3x + 2)(x − 4) > 0.
[x < −2, x > 4
]Ex. 2. (1 − x)(x − 3)(x + 2) < 0.
[− 2 < x < 1, x > 3
]1.2 Rational inequalities
Solve the following inequalities for x ∈ R:
Ex. 3.x2 + x − 2
x2 − 10x + 21<
x − 1x − 3
+ 3x + 1x − 7
.[x < 0, 3 < x < 5, x > 7
]Ex. 4.
x + 12x + 8
−x − 6
x2 + 2x − 48≥
3x − 3x − 6
.[− 8 < x < 6
]Ex. 5.
−9x2− 12x − 4
2x2 − 5x + 2< 0.
[x < −
23, −
23< x <
12, x > 2
]Ex. 6.
(x − a)(x − b)x2 − a2 ≥ 0, a > b > 0.
[x < −a, b ≤ x < a, x > a
]1.3 Irrational inequalities
Solve the following inequalities for x ∈ R:
Ex. 7. 2x − 3 >√
4x2 − 13x + 3.[x ≥ 3
]Ex. 8. x − 8 <
√
x2 − 9x + 14.[x ≤ 2, x ≥ 7
]Ex. 9.
√x − 1 −
√x − 2 < 2.
[x ≥ 2
]Ex. 10.
√x + 2 < 8 +
√x − 6.
[x ≥ 6
]Ex. 11.
√3x − 8 >
√5x + 3 +
√x + 6.
[no solution
]Ex. 12.
√x − 1 ≤ x − 2.
[x ≥
5 +√
52
]
3
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 13.√
x − 1 ≥ −100 − x.[x ≥ 1
]Ex. 14.
√x − 2√
x − 4< 1.
[2 ≤ x < 16
]Ex. 15. 3√
|x + 8| > 1.[x < −9, x > −7
]Ex. 16.
√4 − |x + 3| < 2.
[− 7 ≤ x ≤ 1
]Ex. 17. 3√4 − |x + 3| < 2.
[R]
Ex. 18.√
4 − |x + 2| < 2 − |x|.[−5 +
√17
2< x < 1
]Ex. 19.
√3 − |4x + 2| < 1 − 2|x|.
[0 < x ≤
14
]1.4 Absolute value inequalities
Solve the following inequalities for x ∈ R:
Ex. 20. | |x − 1| − 1 | ≥ 2.[{x ≤ −2} ∪ {x ≥ 4}
]Ex. 21. |x − 2| − |x| < 3.
[R]
Ex. 22. | |x − 2| − |x| | ≤ 3.[R]
Ex. 23. |x2− 2x − 4| ≥ |x| + 2.
[ {x ≤ −2} ∪ {x ≥
3 +√
332
}∪
{3 −√
172
≤ x ≤ 2} ]
Ex. 24. |x − 2| + |x| < 3.[ {−
12< x <
52
} ]Ex. 25.
∣∣∣∣∣x − 2x − 3
∣∣∣∣∣ − |x − 2| < 2.[{x < 1 +
√3} ∪ {x > 2 +
√2}
]1.5 Exponential and logarithmic inequalities
Solve the following inequalities for x ∈ R:
Ex. 26. 4x+163x−2 < 8x.[x <
2 log 3log 108
]Ex. 27. 3 · 52(2x−7)
− 4 · 5(2x−7) + 1 > 0.[x <
72−
log 32 log 5
, x >72
]Ex. 28. log3(2x2
− 7x + 103) > 2.[R]
4
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 29. log5(x2− 7x + 11) < 0.
[2 < x <
7 −√
52
,7 +√
52
< x < 5]
Ex. 30. log10(x + 4)2 > log10(13x + 10).[−
1013< x < 2, x > 3
]Ex. 31. 22x
− 5 · 2x + 4 < 0.[0 < x < 2
]Ex. 32.
62x − 1
+3
2x + 1>
22x − 1
+ 5.[0 < x < 1
]Ex. 33. | log10(3x + 4) − log10 7| < 1.
[−
1110< x < 22
]1.6 Trigonometric inequalities
Solve the following inequalities for x ∈ R:
Ex. 34. 2 sin2 x − cos x − 1 > 0.[π
3+ 2kπ < x < π + 2kπ, π + 2kπ < x <
53π + 2kπ, k ∈ Z
]Ex. 35. cos 2x + 3 sin x ≥ 2.
[π6
+ 2kπ ≤ x ≤56π + 2kπ, k ∈ Z
]Ex. 36. 3 tan2 x − 4
√3 tan x + 3 > 0.
[−π2
+ kπ < x <π6
+ kπ,π3
+ kπ < x <π2
+ kπ, k ∈ Z]
Ex. 37. loga
(12− | sin x|
)< 0, a > 1.[
−16π + 2kπ < x <
16π + 2kπ,
56π + 2kπ < x <
76π + 2kπ, k ∈ Z
]Ex. 38. 3 cos x + sin2 x − 3 > 0.
[not possible
]Ex. 39. 4 cos
(x +
π6
)− 2√
3 cos x + 1 ≥ 0.[−
76π + 2kπ ≤ x ≤
π6
+ 2kπ, k ∈ Z]
Ex. 40.∣∣∣∣∣cos 2x
sin x
∣∣∣∣∣ ≤ 1.[π
6+ 2kπ ≤ x ≤
56π + 2kπ,
76π + 2kπ ≤ x ≤
116π + 2kπ, k ∈ Z
]Ex. 41.
∣∣∣∣∣ tan 2xcot x
∣∣∣∣∣ < 1.[kπ < x <
π6
+ kπ,56π + kπ < x < π + kπ, k ∈ Z
]1.7 Boundedness of numerical sets
Study the boundedness of the following numerical sets, expressing for any of them sup, inf, max
and min by verifying the definition
Ex. 42. A ={ 1
n2 + 1, n ∈N
}.
[inf A = 0, max A = 1
]Ex. 43. A =
{(−1)n
n2 + 2, n ∈N
}.
[min A = −
13, max A =
12
]
5
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 44. A ={x + 2
x − 3, x ∈ R, x > 3
}.
[inf A = 1, sup A = +∞
]Ex. 45. A =
{x + 2x − 2
, x ∈ R, x < 2}.
[inf A = −∞, sup A = 1
]Ex. 46. A =
{ nmn2 + m2 , (n, m) ∈N ×N \ {(0, 0)}
}.
[min A = 0, max A =
12
]Ex. 47. A =
{ nmn2 + m2 , (n, m) ∈N \ {0}
}.
[inf A = 0, max A =
12
]Ex. 48. A =
{n + mn −m
, n,m ∈N, n , m}.
[inf A = −∞, sup A = +∞
]Ex. 49. A =
{ nm
+mn, n,m ∈N \ {0}
}.
[inf A = 2, sup A = +∞
]Study the boundedness of the following numerical sets, expressing for any of them
sup, inf, max and min
Ex. 50. A ={3n + 1
n + 2, n ∈N \ {0}
}.
[min A =
43, sup A = 3
]Ex. 51. A =
{ 11 + 2−n , n ∈N \ {0}
}.
[min A =
23, sup A = 1
]Ex. 52. A =
{ 2nn! + 1
, n ∈N \ {0}}.
[inf A = 0, max A =
43
]Ex. 53. A =
{log n!
n!, n ∈N
}.
[min A = 0, max A = log
√2]
Ex. 54. A =
{n
sin(1 + nπ/2), n ∈N
}.
[inf A = −∞, sup A = +∞
]Ex. 55. A =
√n −√
n + 2n2 , n ∈N \ {0}
. [min A = −
2
1 +√
3, sup A = 0
]Ex. 56. A =
{∣∣∣∣∣(−1)n nn + 3
−15
∣∣∣∣∣ , n ∈N}.
[min A =
15, sup A =
65
]Ex. 57. A =
{∣∣∣∣n2 + sin(nπ2
)∣∣∣∣ , n ∈N
}.
[min A = 0, sup A = +∞
]Ex. 58. A =
{sin
((2n + 1)π
2
)21/(n+1), n ∈N
}.
[min A = −
√2, max A = 2
]Establish if the following numerical sets are bounded; find sup, inf, max and min, if they
exist
Ex. 59. A ={ 1
1 + 2n, n ∈N, n ≥ 1
}.
[inf A = 0, max A =
13
]Ex. 60. A =
{x ∈ R :
xx + 1
>12
}.
[A = (−∞,−1) ∪ (1,+∞); inf A = −∞, sup A = ∞
]
6
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 61. A ={x ∈ R :
√
x2 − 2x <12
x}.
[min A = 2, sup A =
83
]Ex. 62. A = {x ∈ R :
√log(sin x) ∈ R}.
[A = {
π2
+ 2kπ, k ∈ Z}; inf A = −∞, sup A = +∞]
Ex. 63. A = {x ∈ R : 1 ≤ 32x+1 < 9}.[
min A = −12, sup A =
12
]Ex. 64. A =
{x ∈ R : 5 <
15
3x−3≤ 25
}.
[min A =
13, sup A =
23
]Ex. 65. A =
{1 −
(−1)n
n, n ∈N \ {0}
}.
[min A =
12, max A = 2
]
Ex. 66. A =
4
2n + 1, n ∈N, n even
2 −1
n + 1, n ∈N, n odd
.[
inf A = 0, max A = 4]
Ex. 67. Define an infinite set using a non-monotone sequence such that 0 and 1 will be the
inf and sup of the set respectively.
Ex. 68. Find inf e sup of the areas of the surfaces of the rectangles with perimeter equal to
4a, for a a positive real number, different from zero.
1.8 Domain of functions
Determine the domain of the following functions and study the boundedness of such sets.
Then trace a qualitative graph of the functions themselves.
Ex. 69. f (x) =√
x2 − 1.
Ex. 70. f (x) =
√1 − xx + 2
.
Ex. 71. f (x) =4
√|1 − x|x + 2
.
Ex. 72. f (x) = log1/2(1 − |x|).
Ex. 73. f (x) = 6√
log1/3(2 − |x|).
Ex. 74. f (x) =√
log2(x2 − 2x − 5) − 1.
Ex. 75. f (x) =√
log3(2x + 2) − log3 x.
Ex. 76. f (x) =
√log3(
x + 2x
).
7
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 77. f (x) =√
log3(x + 1) − log9(x + 2) + 1.
Ex. 78. f (x) = 2(x+2)/(x2−3x−4).
Ex. 79. f (x) = log5(62x− |4 · 6x
− 1|).
Ex. 80. f (x) = cos(2x − 1
x + 1
).
Ex. 81. f (x) =
√cos
(2x − 1x + 1
).
Ex. 82. f (x) =(cos
(2x − 1x + 1
)−
12
)1/4.
Ex. 83. f (x) =1
sin x + cos x.
Ex. 84. f (x) = 2 log3(sin x + 2 cos x).
Ex. 85. f (x) = log3(sin x + 2 cos x)2.
Ex. 86. f (x) = log23(sin x + 2 cos x).
Ex. 87. f (x) = arccos(x + 1x − 1
).
Ex. 88. f (x) = arcsin( x + 1|x| − 1
).
Ex. 89. f (x) =(log4(sin x)
)1/2.
Ex. 90. f (x) =[2
4√
1−log7(x2+x)− (x2 + x)
]1/2.
Ex. 91. Indicated by D the domain of any function of the exercises in the paragraph 1.8,
determine the set of the interior points D̊ of D and the set of its boundary points ∂D.
Moreover, say if such sets are oper or closed and study their boundedness.
Ex. 92. Determine the set of the images (range) for any function of the exercises in the
paragraph 1.8, and the set of the accumulation points of such sets.
Ex. 93. Given two functions f , g : A ⊆ R→ R, show the following implications:
1. f , g increasing =⇒ f + g increasing;
2. f , g decreasing =⇒ f + g decreasing;
3. f increasing and g strictly increasing =⇒ f + g strictly increasing;
4. f decreasing and g strictly decreasing =⇒ f + g strictly decreasing.
8
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 94. Establish under which conditions the following implication is true:
f , g increasing (or decreasing) =⇒ f · g increasing (or decreasing).
Ex. 95. Furnish an example such that the result of the exercise 94 is, in general i.e. without
further hypothesis, false.
Ex. 96. Show that if f : A ⊆ R→ R is invertible, then
f increasing (decreasing) =⇒ f−1 increasing (decreasing).
Ex. 97. Let f : A ⊆ R→ R be such that 0 < f (A) and increasing. Determine if1f
is increasing
or decreasing.
Ex. 98. Let f , g : A ⊆ R→ R two injective functions. Is the function f + g invertible?
Ex. 99. Let f : X→ Y and g : V →W and let moreover f (X)∩V , ∅. If f and g are invertible
functions, is the composition f ◦ g an invertible function?
Ex. 100. Furnish three different examples of functions f : X→ X such that f ≡ f−1.
1.9 Invertibility of functions
Study the invertibility of the following functions in their natural definition set.
Ex. 101. f (x) = 2x + x.
Ex. 102. f (x) = −x + log1/2 x.
Ex. 103. f (x) = x2 + log3(1 + x).
Ex. 104. f (x) =5x
1 + 5x + x3.
Ex. 105. f (x) = x|x| + 1.
Ex. 106. f (x) =
1
x − 1if x > 1
x + a if x ≤ 1al variare di a ∈ R.
Ex. 107. f (x) =
x2 + ax if x ≤ 0
−1x
if x > 0for any a ∈ R.
Ex. 108. f (x) =
x3 if |x| ≥ 1
ax if |x| < 1.for any a ∈ R.
9
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 109. Let f : X→ Y and g : V →W be two invertible functions such that it is well defined
the composed function g ◦ f . Call f−1 and g−1 their inverses respectively, show that
(g ◦ f )−1 = f−1◦ g−1.
Verify that the following functions are invertible; then determine the inverse of any of
them, specifying its domain.
Ex. 110. f (x) = x|x| + x.
Ex. 111. f (x) = x(x − 2), x ≤ 0.
Ex. 112. f (x) = log1/2(1 − x3).
Ex. 113. f (x) =3x+1
1 + 3x+1.
Ex. 114. f (x) =√
e2x + ex + 1.
Ex. 115. f (x) = sin3(
x2
x2 + 1
), x ≤ 0.
Ex. 116. f (x) = arccos(log2 x).
Ex. 117. f (x) = tan(x3 + 1),π2< x3 + 1 <
32π.
Ex. 118. f (x) = arctan(x3 + 1).
Ex. 119. f (x) = arcsin(√
x2 + 1), x < 0.
10
2 Complex numbers
2.1 Elementary properties of complex numbers
Determine z̄, Im(z) e |z| in the following cases:
Ex. 120. z = 3 − 4i.[3 + 4i, −4, 5
]Ex. 121. z = (2 − i)(−3 + 2i).
[− 4 − 7i, 7,
√65
]Ex. 122. z = (1 − i)(3 − 7i).
[− 4 + 10i, −10, 2
√29
]Ex. 123. z = (2 − 3i)2.
[− 5 + 12i, −12, 13
]Ex. 124. z =
(1 − 2i2
)2.
[−
34
+ i, −1,54
]Ex. 125. z =
23 − i
.[3 − i
5,
15,
√25
]Ex. 126. z =
2 − 3i1 − i
.[5 + i
2, −
12,
√132
]Ex. 127. z = (1 − 2i)3.
[− 11 − 2i, 2, 5
√5]
Ex. 128. z =(1 − i)3
2 − i.
[−2 + 6i5
, −65, 2
√25
]Ex. 129. z =
(1 + 2i)4
(1 − i)2 .[12 +
72
i, −72,
252
]Compute the absolute value and argument of z in the following cases:
Ex. 130. z = 1 + i.[√
2,π4
+ 2kπ, k ∈ Z]
Ex. 131. z = 1 −√
3i.[2,−
π3
+ 2kπ, k ∈ Z]
Ex. 132. z =
√3
3+
i3.
[23,π6
+ 2kπ, k ∈ Z]
Ex. 133. z = i(1 − i).[√
2,π4
+ 2kπ, k ∈ Z]
Ex. 134. z =2
1 +√
3i.
[1, −
π3
+ 2kπ, k ∈ Z]
11
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 135. z =3√
3 + i.
[32, −π6
+ 2kπ, k ∈ Z]
Ex. 136. z = (1 − i)(√
3 + i).[2√
2, −π12
+ 2kπ, k ∈ Z]
Ex. 137. z =1 − i√
3 + i.
[ 1√
2, −
512π + 2kπ, k ∈ Z
]Ex. 138. z =
√3 − i
1 +√
3i.
[1, −
π2
+ 2kπ, k ∈ Z]
Ex. 139. z = (1 − i)12.[64, π + 2kπ, k ∈ Z
]Ex. 140. z =
(1 +
i√
3
)14
.[214
37 ,π3
+ 2kπ, k ∈ Z]
Ex. 141. z =i324− i261
i145 + i492.
[1, −
π2
+ 2kπ, k ∈ Z]
Ex. 142. z =1 − i1039
i2048 − i1457.
[1,π2
+ 2kπ, k ∈ Z]
Ex. 143. z =cos 2θ − i sin 2θ
sinθ + i cosθ.
[1, −θ −
π2
+ 2kπ, k ∈ Z]
Ex. 144. z =12 sin 2θ + i cosθ
1 − i sinθ.[
| cosθ|,π2
se cosθ > 0, −π2
se cosθ < 0, not determinate if cosθ = 0]
2.2 Roots of complex numbers
Compute the following roots of the complex numbers:
Ex. 145.√
1 +√
3i.[±
√3 + i√
2
]Ex. 146.
3√
1 +√
3i.[
3√2(cos
π9
+ i sinπ9
),
3√2(cos
7π9
+ i sin7π9
),
3√2(cos
5π9− i sin
5π9
) ]Ex. 147.
√1 + i1 − i
.[±
1 + i√
2
]Ex. 148. 4
√(1 − i)3 + (1 + i)3.
[1 + i, 1 − i, −1 + i, −1 − i
]Ex. 149.
√2
(1 − i)2
(1 + i)3 .[±
4√2(cos
3π8
+ i sin3π8
)= ±
√√
2 − 1 + i√√
2 + 1√
2
]
12
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Compute the following expressions containing complex numbers:
Ex. 150.
√((1 + i)2
− (1 − i)3)− 2
(3 − i)2 + 6i.
[±
1 + i2
]Ex. 151.
3
√1 +
(2 − 3i)2 + 7i5
.[i, ±√
32−
i2
]Ex. 152. 4
√(1 + 2i)(3 − i)
5−
(1 − i)(1 + 3i)2
.[1 + i√
2,
1 − i√
2,−1 + i√
2,−1 − i√
2
]Ex. 153. 3
√(2 − i)2
− 3(1 + 2i)3 + 11
.[
3√2,−1 ±
√3
3√4
]Ex. 154.
√−1 +
3√i.[0, −2i,
3i2±
√3
2, −
i2±
√3
2
]2.3 Complex equations
2.3.1 Algebraic complex equations
Determine the solutions of the following algebraic equations:
Ex. 155. z2 + 2z + 2 = 0.[z = −1 + i, −1 − i
]Ex. 156. z2
− 6z + 13 = 0.[z = 3 + 2i, 3 − 2i
]Ex. 157. 4z2
− 4z + 17 = 0.[z =
12
+ 2i,12− 2i
]Ex. 158. z3 + 3z2 + z − 5 = 0.
[z = 1, −2 + i, −2 − i
]Ex. 159. z4 + 4 = 0.
[z = 1 + i, 1 − i, −1 + i, −1 − i
]Ex. 160. z2
− iz + 6 = 0.[z = −2i, 3i
]Ex. 161. 4z2
− 2(1 − i)z − i = 0.[z =
12, −
i2
]Ex. 162. 2z2 + iz + 3 = 0.
[z = i, −
3i2
]Ex. 163. 2z2
− 5iz − 2 = 0.[z = 2i,
i2
]Ex. 164. 6z2
− (3 + 2i)z + i = 0.[z =
12,
i3
]Ex. 165. 8z2
− 2(16 + i)z + 5(5 + 2i) = 0.[z = 1 +
i2, 3 −
i4
]Ex. 166. 2z3
− (2 − i)z2 + (1 − i)z − 1 = 0.[z = 1,
i2, −i
]Ex. 167. 2z3 + (2 + 5i)z2 + (3 + 5i)z + 3 = 0.
[z = −1, −3i,
i2
]
13
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
2.3.2 Non-algebraic complex equations
Ex. 168. z|z3| + 16 = 0.
[z = −2
]Ex. 169. z2
|z2| + 16 = 0.
[z = 2i, z = −2i
]Ex. 170. z3
|z| + 16 = 0.[z = −2, z = 1 +
√3i, z = 1 −
√3i]
Ex. 171. z2(1 + |z2|) = −20.
[z = 2i, z = −2i
]Ex. 172. z2(1 − |z2
|) = −20.[z =√
5, z = −√
5]
Ex. 173. z2(4 − |z2|) = 5.
[z =√
5i, z = −√
5i]
Ex. 174. z2(4 − |z2|) = 4.
[z =√
2i, z = −√
2i√
2 + 2√
2i, −√
2 + 2√
2i]
Ex. 175. z2(4 − |z2|) = 3.
[z = 1, z = −1, z =
√3, z = −
√3, z =
√2 +√
7i, −√
2 +√
7i]
Ex. 176.z2
1 + |z2|= −
12.
[z = i, z = −i
]Ex. 177.
z2
1 + |z2|= −2.
[Nessuna soluzione
]Ex. 178.
z2
1 − |z2|= −
12.
[z =
√23
i, z = −
√23
i]
Ex. 179.z2
1 − |z2|= −2.
[z =
i√
3, z = −
i√
3
]Ex. 180.
z4
|z2|== −8.
[z = 2 + 2i, z = 2 − 2i, z = −2 + 2i, z = −2 − 2i
]Ex. 181.
z2
|z4|== −
18.
[z = 2
√2i, z = −2
√2i]
Ex. 182.z4
|z6|== 8.
[z =
1
2√
2, z = −
1
2√
2, z =
i
2√
2, z = −
i
2√
2
]Ex. 183.
z2− |z2|
4 + |z2|+ 1 = 0.
[No solution
]Ex. 184.
z2− |z2|
4 + |z2|+
12
= 0.[z =√
2i, z = −√
2i]
Ex. 185.z2− |z2|
4 + |z2|+
14
= 0.[z = (1 +
√3)i, z = (1 −
√3)i, z = −(1 +
√3)i, z = −(1 −
√3)i
]Ex. 186.
z2− |z2|
4 + |z2|= 0.
[Im(z) = 0
]Ex. 187. z(2 + |z2
|) =3z̄.
[|z| = 1
]
14
3 Limits of one real variable funtions
3.1 Check, using the definition, the following limits
Verify the definition of limit in the following cases:
Ex. 188. limx→1
x = 1.
Ex. 189. limx→+∞
1x
= 0.
Ex. 190. limx→3
(2x + 1) = 7.
Ex. 191. limx→2
x2 = 4.
Ex. 192. limx→0
1x2 = +∞.
Ex. 193. limx→0
1x3 @.
Ex. 194. limx→1
3x = 3.
Ex. 195. limx→π/2
sin x = 1.
Ex. 196. limx→(π/2)−
tan x = +∞.
Ex. 197. limx→1+
x − [x] = 0.
Ex. 198. limx→1−
x − [x] = 1.
Ex. 199. limx→0+
log1/2 x = +∞.
Ex. 200. limx→+∞
x + 22x + 2
=12.
Ex. 201. limx→+∞
sin1x
= 0.
3.2 Computation of limits
Calculate, if they exist real or infinite,the following limits:
Ex. 202. limx→2
(x2 +
1x
).
[92
]
15
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 203. limx→+∞
x + x2
x3 + 1.
[0]
Ex. 204. limx→0
x sin x1 − cos x
.[2]
Ex. 205. limx→+∞
(√x2 + 4 − x
).
[0]
Ex. 206. limx→+∞
(√2x + x2 − x
).
[1]
Ex. 207. limx→+∞
log2(x + x2)log3 x − 1
.[2 log2 3
]Ex. 208. lim
x→0
sin x − xx9/10
.[0]
Ex. 209. limx→4
√
x2 + 1 −√
17x − 4
.[ 4√
17
]Ex. 210. lim
x→0+41/x.
[+∞
]Ex. 211. lim
x→0−41/x.
[0]
Ex. 212. limx→0
sin x −√
x1 − cos 4√x
.[− 2
]Ex. 213. lim
x→0
sin x − x2√
1 − cos x2.
[+∞
]Ex. 214. lim
x→π/22(sin x−1)/x4
.[1]
Ex. 215. limx→0
sin2 x2
+ cos x − 1
x2 .[0]
Ex. 216. limx→+∞
log4
(x + 1x − 1
).
[0]
Ex. 217. limx→0
log3(x + 1)x
.[
log3 e]
Ex. 218. limx→0
log1/2 cos x
x2 .[
log4 e]
Ex. 219. limx→1+
(sin x)1/ log2 x.[0]
Ex. 220. limx→+∞
x3
2x .[0]
Ex. 221. limx→+∞
log3 xx
.[0]
16
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 222. limx→+∞
x3
2log3(log2 x).
[+∞
]Determine domain and image of the following functions, indicating if they are periodic
and even or odd.
Ex. 223. f (x) =√
2 sin2 x + cos x − 1.
Ex. 224. f (x) = log3(sin3 x − cos3 x).
Ex. 225. f (x) = log1/2(| sin 2x| + cos x).
Ex. 226. f (x) = 4(sin x+cos x)/(sin x−cos x).
Ex. 227. f (x) =1
2sin x − 3cos x .
Ex. 228. f (x) = |x|α sin1x3 , al variare di α ∈ R.
Ex. 229. f (x) = arcsin( 2 + ex
e2x − 3
).
Ex. 230. f (x) =4√
tan2(x2 + 1) − tan(x2 + 1) − 6.
Ex. 231. f (x) =5x + 5−x
2.
Ex. 232. f (x) = arctan5x− 5−x
2.
Draw a qualitative graph of the functions studied in the exercises 223, 226, 228, 231 and
232 above.
Calculate, if they exist real or infinite,the following limits:
Ex. 233. limx→+∞
(x + 5)
√x + 1x − 1
− x.[6]
Ex. 234. limx→+∞
x(log(x + 1) − log x).[1]
Ex. 235. limx→+∞
(x3− 2x + 1
x2 + x3
)(2x2+1)/(x−3)
.[e−2
]Ex. 236. lim
x→0
log cos xsin 2x2 .
[−
14
]Ex. 237. lim
x→0+(sin x)x2+3x log x.
[1]
Ex. 238. limx→0
log(1 + sin x)sin 2x + x2 log x
.[12
]
17
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 239. limx→0
e2x−3− e−3
sin x.
[2e−3
]Ex. 240. lim
x→0
sin(√
1 + x2 − 1)
x.
[0]
Ex. 241. limx→0
(log(1 + x) + sin x + x
x + x2
)2
.[9]
Ex. 242. limx→0
e−1/x2+ log
(1 + x1/5
− sin 3√x)
3√x − 2 5√x.
[−
12
]Ex. 243. lim
x→+∞
sin(x5/3x)x42−x .
[0]
Calculate, if they exist real or infinite,the following limits:
Ex. 244. limx→1
√x − cos(x − 1)
log x.
[12
]Ex. 245. lim
x→2
(sin
πx4
)1/ log(3−x).
[1]
Ex. 246. limx→1|x − 1|x−1.
[1]
Ex. 247. limx→0+
x1/ log x.[e]
Ex. 248. limx→+∞
(cos(1/x)cos(2/x)
)(x2+1)/x
.[1]
Ex. 249. limx→0+
(2xx− 1)1/
√x− 1
√x log x
.[2]
Ex. 250. limx→+∞
x2((e1/x + 1)1/2− cos(1/x)).
[+∞
]Ex. 251. lim
x→+∞x2((2e1/x2
− 1)1/2− cos(1/x)).
[32
]Ex. 252. lim
x→3
e−1/(3−x)2+ e(4 − 3 cos(x − 3))1/5
− e√
4−x√1 − cos(x − 3)
.[−
e√
2
]Ex. 253. lim
x→+∞sin(1/x) · log(x2 + e1/x + 2x2/(x+1)).
[log 2
]
Ex. 254. limx→+∞
1
log10(x2 + x + 1)
sin1
xx + 1
log10(x3 + x + 1)
−1
.[(32
)10]
Ex. 255. limx→0
arcsin√
x√cos 4√x − 1
.[− 4
]
18
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 256. limx→0
(1 + sin x)1/ arctan x.[e]
Calculate, if they exist real or infinite,the following limits:
Ex. 257. limx→+∞
x2 + sin x
x + log(x + e2x2
) . [12
]Ex. 258. lim
x→+∞
(x4e−x + sin(1/x2) + 1
)√1+2x4
.[e√
2]
Ex. 259. limx→+∞
(√x + x3 − x
)log
( √4x + 1
2√
x + 3
)x arctan x
.[−
3π
]Ex. 260. lim
x→+∞
xarctan x− xπ/2
(1 + x)π/2+1/√
log x.
[0]
Ex. 261. limx→+∞
xarctan x− xπ/2
(1 + x)π/2−1.
[−∞
]
Ex. 262. limx→0+
e−1/x + x2 +1
log2 x+ x log
(e−1/x + e−2/x
)+ 1
ex − 1.
[+∞
]Ex. 263. lim
x→0+
x sin x − cos x + ex2/2√
1 − cos x · arcsin x.
[ 4√
2
]Ex. 264. lim
x→1
(x2− 2x + 1) tan(x − 1) − sin3(x − 1)√
cos(x − 1) − 1.
[0]
Ex. 265. limx→0+
x(cos√
x3 − 1)
+ sin2 x3/4
x3e−1/√
x +√
x(ex2− 1
) .[
+∞]
Ex. 266. limx→0+
x(cos√
x3 − 1)
+ sin2 x3/4
x3e−1/√
x +(ex2− 1
)/√
x.
[1]
Ex. 267. limx→0+
log | log x| + log x
log(1 + xlog x
) .[0]
Ex. 268. limx→1
e3x−x2− e2 cos(x − 1) − x + 1log (sinπx/2)
.[(2eπ
)2]
Calculate, if they exist real or infinite,the following limits:
Ex. 269. limx→0+
xx; limx→0+
xxx; lim
x→0+xxxx
.[1; 0; 1
]
Ex. 270. limx→0+
n times︷︸︸︷xx.
..x
.[1 if n is even, 0 if n is odd
]
19
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 271. limx→1
((x − 1)2
sin(πx)(e − ex))log x.
[1]
Ex. 272. limx→+∞
cos(1/x) − e−1/x2
(√x4 − x2 − x2
)log
√x2 + 2x2 + 1
.[− 2
]
Ex. 273. limx→0+
sin(ex2− cos x + 2 sin x2
√1 + 2 sin2 x)
2 sin2 x.
[74
]
Ex. 274. limx→0+
√ex2− cos x + 2 sin x3
√
1 + 2 sin x3
2 sin3 x.
[+∞
]Ex. 275. lim
x→0+
√1 + x sin x −
√cos 2x
tan2(x/2).
[6]
Ex. 276. limx→0
log(2 − cos x)(2 − cos x)1/x2sin x2
sin2 x2.
[ √e2
]Ex. 277. lim
x→0
log(cos2 x)(x −√
x2 + 3x + 1)1 + e−1/x − cos x
.[2]
Ex. 278. limx→0
((sin x + 2)2 log(sin x + 1)
)(√
1+3x2−1)/x2
.[0]
Ex. 279. limx→+∞
log(
e−1/x
x4+ 1
)+ sin3(1/x)
log(
2 + x3
x3
) .[12
]Arrange in growing order of infinity (infinitesimal) the following functions and sequences,
after having determined the order of infinite (infinitesimal), if it exits as a real number.
Ex. 280. For x→ +∞: a)ex
x2 , b) x log x, c)x2
log x, d)
1sin(1/x)
.[d, b, c, a. ord d=1
]Ex. 281. For n→ +∞: a) 2n, b) n!, c) nn, d)
(32
)n2
.[a, d, b, c
]Ex. 282. For x→ +∞: a) xx, b) x log2 x, c) x2 log x, d)
x5 + x3 + 2x2 + 1
logx + 1
x.[
b, d, c, a. ord d=2]
Ex. 283. For x→ 0+: a)1
log x, b) x2, c)
3√1 − cos x√
arcsin x, d) log x · arcsin x.[
a, c, d, b. ord b=2, c=16
]Ex. 284. For x→ 0+: a) log x, b) log | log x|, c)
1x log x
, d)1
log(1 + x).[b, a, c, d. ord d=1
]
20
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 285. For x→ 1+: a) e−1/(x−1)2, b) 10√x − cos(x − 1), c) sin3 3√
x2 − x, d)x − 1
log20(x − 1).[
c, d, b, a. ord b=1, c=13
]Ex. 286. For x→ 2+: a)
1(x − 2)3/2
, b)1
(x − 2)3/4 log(x − 2), c) e
√x−2/ sin(x−2),
d) (x − 2)1/(2−x).[a, b, c, d. ord a= 3
2
]Ex. 287. For x→ 0+: a) x arctan x, b)
1 − cos xlog x
, c) xx− 1, d) sin3 4√x.[
d, c, b, a. ord a=2, d=34
]Arrange in growing order of infinity (infinitesimal) the following functions and sequences,
after having determined the order of infinite (infinitesimal), if it exits as a real number.
Ex. 288. For x→ +∞: a) x2, b) log(1 + x3 + ex3), c)
x2
x + 1, d)
(x2
x + 1
)1+1/√
log x
.[c, d, a, b. ord a=2, b=3, c=1
]Ex. 289. For n→ +∞: a)
√n
n2 + 1, b)
1n log n
, c)log2 n
n, d)
n!(n + 1)! − (n − 1)!
.[c, d, b, a. ord a= 3
2 , d=1]
Ex. 290. For n→ +∞: a) ( n√n− 1)−1, b) n(√
3 + n2 −n), c) (cos(1/n)− 1) · 2n3/(n+1), d) nn.[a, b, c, d. ord b=2
]Ex. 291. For x→ 0+: a)
x2(1 − cos x)2
log(1 + sin4 x), b) log(x+1), c) x log x, d) sin(x log(1+x)) ·log x.[
c, b, d, a. ord a=2, b=1]
Ex. 292. For x→ +∞: a)x2 log(2 − cos(1/x))
sin2(1/x), b)
x√
x
x100, c) x2 log
(x2 + 1
x
),
d) x log100(1 + x).[d, a, c, b. ord a=2
]Ex. 293. For x→ 3+: a) (e
(x−3)2
(3−x)(x+1)3 − 1) sin(x − 3)9/4, b) sin3(x − 3), c) (x − 3)3 log(x − 2),
d) (x − 3)3 log10(x − 3).[d, b, a, c. ord a= 13
4 , b=3, c=4]
Ex. 294. For x→ 0+: a) x log(1 + x2), b) x2−x/(x2+1), c)
3√
x2 + x4√
x2 + 2x
25
, d) x3 log10 x.[b, c, d, a. ord a=3, b=2, c=4, c= 25
12
]Ex. 295. For x→ 0+: a) x arctan
√x, b)
(1 − cos x)2√
x + 1√
x4 + 1 log(1 + x2), c) x2 log
(x2 + 1
x
)e√
x,
d) sin(x3 log x).[a, c, b, d. ord a=3
2 , b=2]
21
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Calculate the limit of the following sequences:
Ex. 296. limn→+∞
en2
nn .[
+∞]
Ex. 297. limn→+∞
en3/2
nn2 + en.
[0]
Ex. 298. limn→+∞
√
n2 + n3 − n + sin n4√
1 + n5 + 2n6.
[ 4√22
]Ex. 299. lim
n→+∞
2(1+log1/2 n)
n1/2.
[0]
Ex. 300. limn→+∞
(log(n2 + 1) − log n − log(n + 1))√
1 + n2.[− 1
]Ex. 301. lim
n→+∞
2n− 3n
4n .[0]
Calculate the limit of the following sequences:
Ex. 302. limn→+∞
n√
(n2 + 1) sin(1/n).[1]
Ex. 303. limn→+∞
nn
(n!)!.
[0]
Ex. 304. limn→+∞
n√
n!.[
+∞]
Ex. 305. limn→+∞
∣∣∣∣logn
n + 1
∣∣∣∣ 1−2√
n+1n+√
n .[1]
Ex. 306. limn→+∞
(1 +
n!nn
) (n−1)nn(n+1)!
.[
e√e]
Ex. 307. limn→+∞
√
n2 + 1 arcsin(e−n +
1n2 + n
).
[0]
Ex. 308. limn→+∞
(1 + cos(1/n) − cos(2/n))−(arcsin(1/n))n2
.[1]
Ex. 309. limn→+∞
(n2 + n3 + 3√
n + n + n3 − 1e−1/n)n.
[1]
Ex. 310. limn→+∞
n6 + en log n + 2n4 arcsin(1/n)
nn − n! + en3 .[1]
Ex. 311. limn→+∞
n2√n3 + 1 + en2 .
[e]
Ex. 312. limn→+∞
n√
en + sin(πn/2).[e]
Ex. 313. limn→+∞
n√
2 + sin n.[@]
22
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
*Ex. 314. Let {an} be a positive terms sequence such that
limn→+∞
logan
an+1≥ 0.
Give at least two counterexamples showing that from this relation is not possible deduce that
limn→+∞
an = +∞
Moreover, say under which further hypothesis the result would be true.
Ex. 315. Using the comparison theorem, show that
limn→+∞
1n2 + 1
+1
n2 + 2+ · · · +
1n2 + n
= 0.
*Ex. 316. Let {an} be a positive terms sequence. Show that
limn→+∞
an+1
an= r ≥ 0 ⇒ lim
n→+∞
n√an = r
Use the sequence an = e1/n + sin(πn/2) + 1 to show that in general the converse is not true.
*Ex. 317. Show, exhibiting a counterexample, that if {an} is a non-negative terms sequence,
then
limn→+∞
a1/nn = l ; lim
n→+∞
an
ln= 1
Moreover, show that if limn→∞
a1/nn = l > 1 then an → +∞ for n→ +∞.
23
4 Study of functions of one real variable
4.1 Asymptotes
Determine the possible asymptotes (vertical, horizontal, oblique) for the following functions,
after having indicated their domain. Moreover, calculate the limit of the functions to the
boundary points of their domain.
Ex. 318. f (x) =x + 1
3 − 2x.
Ex. 319. f (x) =1
x(x − 2).
Ex. 320. f (x) =
√
x4 + 1x − 2
.
Ex. 321. f (x) = x log(1 + x).
Ex. 322. f (x) =x
x2 + 1.
Ex. 323. f (x) =x
x2 − 1.
Ex. 324. f (x) = x arcsin1
x + 1.
Ex. 325. f (x) = elog2(x/(x−1))+log(3x−3)+2.
Ex. 326. f (x) = log(1 − 3ex + 2e2x).
Ex. 327. f (x) = x ex/(x2−1).
Ex. 328. f (x) = x√
cosx
x2 + 1.
Ex. 329. f (x) =log |x|
3 + log |x|+√
x2 + 2x.
Ex. 330. f (x) = x arctan x. (Use the formula arctan x + arctan1x
=π2, x > 0).
Ex. 331. f (x) = x1+1/ log x.
Ex. 332. f (x) = x1+log x/√
1+log2 x.
Ex. 333. f (x) =x2
x4 − 1e−1/x2
.
24
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
4.2 Continuity and derivability
Determine the domain and the set of continuity of the following functions.
Ex. 334.
f (x) =
x − [x] − 1, x ≤ 2
x − [x], x > 2.
Ex. 335. f (x) = [x] +√
x − [x].
Ex. 336. f (x) = 41/ sin x.
Ex. 337. f (x) =sin(log x)
log x.
Ex. 338.
f (x) =
sin(cot x), x , kπ, k ∈ Z
0, x = kπ, k ∈ Z.
Ex. 339. Determine a ∈ R such that the following function result to be continuous
f (x) =
x2− 1
x + 1, x , −1
a, x = −1.
Ex. 340. Say if it is possible to apply the Weierstass theorem about the existence of the
extremes to the following function
f (x) =
x, 0 ≤ x < 1
1 − x, 1 ≤ x ≤ 3.
Determine the set of continuity and the set of derivability of the following functions and
calculate their derivative.
Ex. 341. f (x) = tan 2x.
Ex. 342. f (x) = e2x− e−2x.
Ex. 343. f (x) = 32x.
Ex. 344. f (x) = xx2+1.
Ex. 345. f (x) =2x + 3x − 4
.
Ex. 346. f (x) =
√2x + 3x − 4
.
Ex. 347. f (x) =
√x
x2 + 1.
25
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 348. f (x) = (arcsin x)3.
Ex. 349. f (x) = esin x.
Ex. 350. f (x) = arctan( x1 − x2
).
Ex. 351. f (x) = log tan x.
Ex. 352. f (x) = arcsin(
11 +√
x
).
Ex. 353. f (x) = arcsin(
x2
x2 − 1
).
Ex. 354. f (x) = x e1/(1−x).
Ex. 355. f (x) = 2arccos 3x.
Ex. 356. f (x) = log 2|x|.
Ex. 357. f (x) =log x
3 − 2 log(2x).
Ex. 358. f (x) = |x|x + ex.
Ex. 359. f (x) =√
x2 + x4 arctan x.
Ex. 360. f (x) =√
1 − cos x.
Ex. 361. f (x) =
√log
(x2
x2 − 1
).
The same work (determination of continuity, derivability and calculation of the derivative)
is recommended also for the functions in the exercises in paragraphs 1.8 and 4.1.
4.3 Invertibility and derivative of the inverse function
Verify the invertibility of the following functions and determine the domain of derivability
of the respective inverse functions.
Ex. 362. f (x) = 2x + log x.
Ex. 363. f (x) = −x + e−2x.
Ex. 364. f (x) = x|x| + log(1 + x).
Ex. 365. f (x) = x + sin x.
26
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 366. f (x) = x√|x| + arctan x.
Ex. 367. f (x) =5√1 − x − cos x.
Ex. 368. For any of the following function f (x), determine:
f ′(2), f ′(1), f ′(1 + log 2), f ′(π2
+ 1), f ′(1 +π4
), f ′(0).
Moreover, write the equation of the tangent line passing for the point indicated.
Ex. 369. Use the mean value theorem to show that
| sin x − sin y| ≤ |x − y|, x, y ∈ R .
4.4 Critical points
Determine the possible critical points for the following functions.
Ex. 370. f (x) =x
x2 + 1.
Ex. 371. f (x) =x
x2 − 1.
Ex. 372. f (x) =log x
x.
Ex. 373. f (x) = xe−1/x.
Ex. 374. f (x) =√
x∣∣∣∣∣1 +
1log x
∣∣∣∣∣ .Ex. 375. f (x) = x log x.
Ex. 376. f (x) = x3 + x2− x.
Ex. 377. f (x) =√−x(x + 1).
Ex. 378. f (x) = ex(32|x| +
12
(3x − 8)).
Ex. 379. f (x) = ((2 − x)6)log |x−2|.
4.5 Derivability and Monotony
Determine the intervals of monotony for the functions in the paragraph 4.4.
27
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
4.6 Taylor and Mac Laurin Polynomials
Determine the Mac Laurin polynomial of the following functions to the indicated order.
Ex. 380. f (x) = sin(x2), to the order 4.
Ex. 381. f (x) =√
1 + 2x, to the order 3.
Ex. 382. f (x) = log(1 + x3), to the order 8.
Ex. 383. f (x) = sin2 x, to the order 4.
Ex. 384. f (x) = ex+1, to the order 5.
Determine the Taylor polynomial, centered in x0 and to the indicated order, for the
following functions.
Ex. 385. f (x) = ex, x0 = 2, to the order 3.
Ex. 386. f (x) = cos x, x0 = 3, to the order 4.
Ex. 387. f (x) = log(1 + x), x0 = 2, to the order 3.
Ex. 388. Determine the Mac Laurin polynomial of order 4, for the function
f (x) = log(1 + x sin x) .
Determine the Mac Laurin polynomial of order 5, for the following functions.
Ex. 389. f (x) = (1 + x)ex.
Ex. 390. f (x) = x sin x + cos x.
Ex. 391. f (x) = sin x · log(1 + x).
4.7 Using Taylor polynomials for the calculation of limits
Calculate the following limits.
Ex. 392. limx→+∞
x3
x + 1
(e1/(x+1)
− 1)− x.
[−
32
]Ex. 393. lim
x→0+
12
x sin x + cos x − ex4
x2 log(1 + x2).
[−
2524
]Ex. 394. lim
x→1+
e−1/(x+1) +√
xx − x log x(x log (x cos(x − 1))
)2 .[
+∞]
Ex. 395. limx→+∞
x5 + x2 log x
x3 + x6 log(2 arctan x
π
)−
2π
x5
[−π4
]
28
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
4.8 Uniform Continuity
Ex. 396. Verify, using the definition, that f (x) = x2 is not an uniform continuous function
over X = [1, +∞).
Ex. 397. Establish if f (x) =arctan x
xis a uniform continuous function over the following
domains:
D1 = (0, +∞); D2 = (1, +∞); D3 = [1, +∞); D4 = (−∞, −1) ∪ (2, +∞).
Ex. 398. Verify if f (x) = x − log x results to be a Lipschitz function over the domain
D = [1, +∞).
Verify if the following functions result to be uniformly continuous over their domain:
Ex. 399.
f (x) =
xe−1/|x|, if x , 0,
0, if x = 0.
Ex. 400.
f (x) =
2 sin x + 1, if x < 0,
log(e(2x + 1)), if x ≥ 0.
Ex. 401. f (x) = sin(esin x)
For any of the following functions determine a ∈ R such that they result to be continuous.
Then check if, for such an a, the functions result to be also uniformly continuous throughout
their domain of definition.
Ex. 402.
f (x) =
a(ex− 1), if x < 1,
e−x, if x ≥ 1.
Ex. 403.
f (x) =
log x
x+ e1−x, if x > 1,
a, if x = 1,√
2 − x +π4− arctan x, if x < 1.
Ex. 404.
f (x) =
√
x2 − 2x + 2, if x ≤ 0,a log(x + 1)
x, if x > 0.
29
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 405.
f (x) =
2 sin x2−
1√log(1 + x) + 1
, if x > 0,
a, if x = 0,
x(ex + 1) − 1, if x < 0.
30
5 Integrals of one-variable functions and numerical series
5.1 Immediate indefinite integrals (primitives)
Calculate the following indefinite integrals (primitives).
Ex. 406.∫
14√
x3dx.
[4x1/4 + c
]Ex. 407.
∫ √3qx dx, q ∈ R+.
[23
(3q)1/2x3/2 + c]
Ex. 408.∫
(a2/3− x2/3)3 dx, a ∈ R.
[a2−
95
a4/3x5/3 +97
a2/3x7/3−
13
x3 + c]
Ex. 409.∫
Pn(x) dx, Pn(x) =
n∑k=0
akxk, ak ∈ R.[ n∑
k=0
ak
k + 1xk+1 + c
]
Ex. 410.∫ n∑
k=0
αkeβkx dx, αk, βk ∈ R, βk , 0.[ n∑
k=0
αk
βkeβkx + c
]
Ex. 411.∫ n∑
k=0
αk sin βkx dx, αk, βk ∈ R, βk , 0.[−
n∑k=0
αk
βkcos βkx + c
]Ex. 412.
∫x2− 3x + 1
xdx.
[12
x2− 3x + log |x| + c
]Ex. 413.
∫3 +√
x5√
x2dx.
[5
5√
x3 +1011
10√
x11 + c]
Ex. 414.∫
a +√
1 − x2√
1 − x2dx, a ∈ R.
[a arcsin x + x + c
]Ex. 415.
∫x2
1 + x2 dx.[x − arctan x + c
]Ex. 416.
∫tan2 x dx.
[tan x − x + c
]Ex. 417.
∫cot2 x dx.
[− cot x − x + c
]Ex. 418.
∫1 + 2x2
x2(1 + x2)dx.
[−
1x
+ arctan x + c]
31
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 419.∫
sin 2xcos x
dx.[− 2 cos x + c
]Ex. 420.
∫x5 + 1x + 1
dx.[x5
5−
x4
4+
x3
3−
x2
2+ x + c
]Ex. 421.
∫xn− an
x − adx, a ∈ R.
[xn
n+ a
xn−1
n − 1+ a2 xn−2
n − 2+ · · · + an−1x + c
]Ex. 422.
∫dx
sin2 x cos2 x.
[tan x − cot x + c
]Ex. 423.
∫cos 2x
sin x + cos xdx.
[cos x + sin x + c
]Ex. 424.
∫sin2 x
2dx.
[12
(x − sin x) + c]
Ex. 425.∫
cos2 x3
dx.[12
x +34
sin2x3
+ c]
Ex. 426.∫
1
sin2 x2
cos2 x2
dx.[2 tan
x2− 2 cot
x2
+ c]
5.2 Indefinite integrals by substitution
Calculate the following indefinite integrals using, for instance, the method of substitution of
variable.
Ex. 427.∫√
sin x cos x dx.[23
sin3/2 x + c]
Ex. 428.∫
x1 − x2 dx.
[−
12
log∣∣∣1 − x2
∣∣∣ + c]
Ex. 429.∫
1a2 + x2 dx, a ∈ R+.
[1a
arctanxa
+ c]
Ex. 430.∫
1a2 − x2 dx, a ∈ R+.
[−
12a
log∣∣∣∣a − xa + x
∣∣∣∣ + c]
Ex. 431.∫ √
a − x√
a + xdx, a ∈ R+.
[a arcsin
xa
+12
√
a2 − x2 + c]
Ex. 432.∫
1 + e−x
1 + xe−x dx.[
log |x + ex| + c
]Ex. 433.
∫1
√
a − bx2dx, a, b ∈ R+
[ 1√
barcsin
√ba
x + c]
Ex. 434.∫
x√
1 − x2dx.
[−
√
1 − x2 + c]
32
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 435.∫
1
x√
5x − 7dx.
[ 2√
7arctan
√5x + 7
7+ c
]Ex. 436.
∫sinα x cos x dx, α , −1.
[ 1α + 1
sinα+1 x + c]
Ex. 437.∫
1e−x + ex dx.
[arctan ex + c
]Ex. 438.
∫cos(log x)
xdx.
[sin(log x) + c
]Ex. 439.
∫ √x
1 + xdx.
[2(√
x − arctan√
x) + c]
Ex. 440.∫
x2
(x − 1)3 dx.[
log |x − 1| −2
x − 1−
12(x − 1)2 + c
]Ex. 441.
∫x
√
a4 − x4dx, a , 0.
[12
arcsinx2
a2 + c]
Ex. 442.∫
cot xsinα x
dx, α ∈ R+.[−
1α sinα x
+ c]
Ex. 443.∫
1
x√
x2 − a2dx, a , 0.
[−
1a
arctanax
+ c]
Ex. 444.∫ √
x2 − a2
xdx, a ∈ R.
[√x2 − a2 − a arccos
ax
+ c]
Ex. 445.∫
ax + bcx + d
dx, a, b, c, d ∈ R, c , 0.[ 1c2
(a(cx + d) + (bc − ad) log |cx + d|
)+ cons.
]Ex. 446.
∫e2x√
ex − 1dx.
[23
√(ex − 1)3 + 2
√
ex + 1 + c]
Ex. 447.∫
tan xlog(cos x)
dx.[− log
∣∣∣log(cos x)∣∣∣ + c
]Ex. 448.
∫1
(a + x)(a2 − x2)1/2dx, a , 0.
[−
1a
√a − xa + x
+ c]
Ex. 449.∫
1sin x cos x
dx.[
log | tan x| + c]
Ex. 450.∫
1sin x
dx.[
log |sin x
1 + cos x| + c
]Ex. 451.
∫1
cos xdx.
[log |
1 + sin xcos x
| + c]
Ex. 452.∫
1√
x(a + x)dx, a ∈ R+
[ 2√
aarctan
√xa
+ c]
33
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 453.∫
1√
a2 + x2dx, a ∈ R.
[log
∣∣∣∣√a2 + x2 + x∣∣∣∣ + c
]Ex. 454.
∫1
(a2 + x2)3/2dx, a , 0.
[ 1a2
x√
a2 + x2+ c
]Ex. 455.
∫1
(a2 + x2)5/2dx, a , 0.
[ 1a4
(x
√
a2 + x2−
13
x3√(a2 + x2)3
) + c]
Ex. 456.∫
1(a2 + x2)7/2
dx, a , 0.[ 1a6 (
x√
a2 + x2−
23
x3√(a2 + x2)3
+15
x5√(a2 + x2)5
) + c]
*Ex. 457.∫
1(a2 + x2)(2n+1)/2
dx, a , 0,n ∈N.[use the results of the previous exercises
]5.3 Indefinite integrals by parts
Calculate the following indefinite integrals using, for instance, the method of integration by
parts.
Ex. 458.∫
log xx3 dx.
[−
12x2
(log x +
12
)+ c
]Ex. 459.
∫sin3 x dx.
[−
13
(sin2 x cos x + 2 cos x
)+ c
]Ex. 460.
∫sin4 x dx.
[38
x −14
sin2 x +1
32sin 4x + c
]Ex. 461.
∫sin5 x dx.
[− cos x +
23
cos3 x −15
cos5 x + c]
Ex. 462.∫
sin xex dx.
[−
12(sin xe−x + cos xe−x) + c
]Ex. 463.
∫x3 arctan x dx.
[x4
4arctan x +
12
(x −
x3
3− arctan x
)+ c
]Ex. 464.
[ logα+1 xα + 1
+ c if α , 0, 1; log |x| + c, if α = 0; log | log x| + c if α = −1]
Ex. 465.∫
xex dx.[xex− ex + c
]Ex. 466.
∫x2ex dx.
[ex(x2
− 2x + 2) + c]
Ex. 467.∫
xnex dx, n ∈N.[ex(xn
− nxn−1 + n(n − 1)xn−2− · · · + (−1)nn!) + c
]Ex. 468.
∫x sin x dx.
[− x cos x + sin x + c
]
34
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 469.∫
x cos x dx.[x sin x + cos x + c
]Ex. 470.
∫x2 sin x dx.
[− x2 cos x + 2x sin x + 2 cos x + c
]Ex. 471.
∫x2 cos x dx.
[− x2 cos x + 2x sin x + 2 cos x + c
]Ex. 472. In =
∫xn sin x dx, n ∈N,n > 1.
[Let I1 =
∫x cos x dx, In = −xn cos x + nIn−1 =
= −xn cos x + n(−xn−1 cos x + (n − 1)(−xn−2 cos x + . . . + 2∫
x cos x dx) . . .)]
Ex. 473. In =
∫xn cos x dx, n ∈N,n > 1.
[Let I1 =
∫x sin x dx, In = xn sin x − nIn−1 =
= xn sin x − n(xn−1 sin x − (n − 1)(xn−2 sin x − . . . − 2∫
x sin x dx) . . .)]
Ex. 474.∫
x sin2 x dx.[12
(−x sin x cos x +
x2
2+
sin2 x2
)+ c
]Ex. 475.
∫√
1 − x2 dx.[12
(x√
1 − x2 + arcsin x)
+ c]
Ex. 476.∫
x arcsin x dx.[x2
2arcsin x +
14
(x√
1 − x2 − arcsin x)
+ c]
Ex. 477.∫
xcos2 x
dx.[x tan x + log | cos x| + c
]Ex. 478.
∫arcsin2 x dx.
[x arcsin2 x + 2
√
1 − x2 arcsin x − 2x + c]
Ex. 479.∫
earcsin x dx.[12
earcsin x(x +√
1 − x2)
+ c]
Ex. 480.∫
sin px cos qx dx, p, q ∈ R, p , q.[ qq2 − p2 sin px sin qx +
pq2 − p2 cos px cos qx + c
]Ex. 481.
∫√
x2 + a dx, a ∈ R.[12
(x√
x2 + a + a log∣∣∣∣√x2 + a + x
∣∣∣∣ + c]
Ex. 482.∫
ex sin x dx.[12
ex(sin x − cos x) + c]
Ex. 483.∫
ex cos x dx.[12
ex(sin x + cos x) + c]
Ex. 484.∫
eαx sin x dx, α ∈ R.[ 1α2 + 1
eαx(α sin x − cos x) + c]
Ex. 485.∫
eαx cos x dx, α ∈ R.[ 1α2 + 1
eαx(sin x + α cos x) + c]
35
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 486.∫
eαx sin βx dx, (α, β) ∈ R2, (α, β) , (0, 0).[ 1α2 + β2 eαx(α sin βx − β cos βx) + c
]Ex. 487.
∫eαx cos βx dx, (α, β) ∈ R2, (α, β) , (0, 0).
[ 1α2 + β2 eαx(β sin βx + α cos βx) + c
]*Ex. 488.
∫ex cosn x dx, n ∈N.
(Use the formula:
cosn x =
1
2n−1
bn/2c∑k=0
(nk
)cos(n − 2k)x, n odd
12n
(n
n/2
)+
12n−1
bn/2c−1∑k=0
(nk
)cos(n − 2k)x, n even
and the result of the exercise 487.)
*Ex. 489.∫
ex sinn x dx, n ∈N.
(Use the formula:
sinn x =
1
2n−1
bn/2c∑k=0
(−1)bn/2−kc(nk
)sin(n − 2k)x, n odd
12n
(n
n/2
)+
12n−1
bn/2c−1∑k=0
(−1)bn/2−kc(nk
)sin(n − 2k)x, n even
and the result of the exercise 486.)
*Ex. 490. Im,n =
∫sinm x cosn x dx, m,n ∈ Z.
(One obtains the following equivalent reduction formulas:
Im,n = −sinm−1 x cosn+1 x
n + 1+
m − 1n + 1
Im−2,n+2 =sinm+1 x cosn−1 x
m + 1+
n − 1m + 1
Im+2,n−2 =
=sinm+1 x cosn+1 x
m + 1+
m + n + 2m + 1
Im+2,n = −sinm+1 x cosn+1 x
n + 1+
m + n + 2n + 1
Im,n+2)
*Ex. 491.∫
eαx sinm βx dx, α, β ∈ R, m ∈ Z.
(Use the results of the previous exercises)
*Ex. 492.∫
eαx cosn βx dx, α, β ∈ R, n ∈ Z.
(Use the results of the previous exercises)
5.4 Determine the following indefinite integrals (primitives).
Ex. 493.∫
x2 + 2(x − 3)2(x + 2)
dx.[1925
log |x − 3| −11
5(x − 3)+
625
log |x + 2| + c]
36
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 494.∫
4x − 3(x − 1)(x − 2)3 dx.
[− log |x − 1| + log |x − 2| +
2(x − 2)
−5
2(x − 2)+ c
]Ex. 495.
∫x5 + x4
− 8x3 − 4x
dx.[x3
3+
x2
2+ 4x + 2 log |x| + 5 log |x − 2| − 3 log |x + 2| + c
]Ex. 496.
∫x
(x2 + 1)(x − 1)dx.
[12
(−
12
log(x2 + 1) −12
arctan x + log |x − 1|)
+ c]
Ex. 497.∫
x + 1x2 + 1
dx.[12
log(x2 + 1) + arctan x + c]
Ex. 498.∫
x3− 6
x4 + 6x2 + 8dx.
[−
52(x − 2)2 +
1x − 2
+ log |x − 2| − log |x − 1| + c]
Ex. 499.∫
x3− 2x2 + 5
x4 + 3x3 + 3x2 − 3x − 4dx.[1
4log |x − 1| −
12
log |x + 1| +58
log |x2 + 3x + 4| −31√
78
arctan2x + 3√
7+ c
]Ex. 500.
∫2x3− 3x + 3
(x − 1)(x2 − 2x + 5)dx.[11
4log
(|x2− 2x + 5|
)+
12
log (|x − 1|) −52
arctan(14
(2x − 2))
+ 2x + c]
Ex. 501.∫
x2 + x + 1/2x2 + 1
dx.[12
log(x2 + 1) −12
arctan(x) + x + c]
Ex. 502.∫
3x2− 6x + 7
(x − 2)2(x + 5)dx.
[167
log (|x + 5|) +57
log (|x − 2|) −1
x − 2+ c
]Ex. 503.
∫2x2 + x
(x2 + 1)(x2 + 2x + 2)dx.[−
12
log(|x2 + 2x + 2|
)+
12
log(x2 + 1) + arctan(12
(2x + 2))
+ c]
Ex. 504.∫
x3 + x − 1(x2 + 2)2 dx.
[12
log(x2 + 2) −1
252
arctan(x√
2) −
x − 24x2 + 8
+ c]
Ex. 505.∫
1(x3 + 1)2 dx.[1
9log
(|x2− x + 1|
)+
29
log (|x + 1|) +2
332
arctan(2x −1√
3) +
x3x3 + 3
+ c]
Ex. 506.∫
1(x2 + 1)2 dx.
[12
arctan x +x
2x2 + 2+ c
]Ex. 507.
∫4
x4 + 1dx.[
4(
1
252
log(|x2 +
√
2x + 1|)−
1
252
log(|x2−
√
2x + 1|)
+1
232
arctan(
2x +√
2√
2
)+
1
232
arctan(
2x −√
2√
2
))+ c
]
37
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 508.∫
tan2 xtan3 x + 1
dx.
[16
log(| tan2 x − tan x + 1|
)+
16
log (| tan x + 1|) −14
log(tan2 x + 1
)+
arctan(
2 tan x − 1√
3
)√
3−
12
x + c]
Ex. 509.∫
sin2 xcos2 x + 2 sin2 x
dx.[x −
arctan(√
2 tan x)
√2
+ c]
Ex. 510.∫
1sinm x cosn x
dx, m,n ∈N.[]
Ex. 511.∫
cos mx sin nx dx, m,n ∈N.[]
Ex. 512.∫ √
x4√x + 1
dx.[20
(x
14 + 1
)+
45
(x
14 + 1
)5− 5
(x
14 + 1
)4+
403
(x
14 + 1
)3− 20
(x
14 + 1
)2− 4 log
(x
14 + 1
)+ c
]Ex. 513.
∫ 3√x√
x + x2dx. []
Ex. 514.∫
1 + tan x1 − tan x
dx.[
12 log
(tan2 x + 1
)− log (| tan x − 1|) + c
]Ex. 515.
∫1
3 + 5 cos xdx.
[2(18
log(∣∣∣∣∣ sin x
cos x + 1+ 2
∣∣∣∣∣) − 18
log(∣∣∣∣∣ sin x
cos x + 1− 2
∣∣∣∣∣)) + c]
38
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
5.5 Definite Integrals
Determine the following definite integrals:
Ex. 516.∫ 3
−2
xx2 + 1
dx.[ log 2
2
]Ex. 517.
∫ 3
−3
xx2 + 1
dx.[0]
Ex. 518.∫ 3
−3
x2
x2 + 1dx.
[6 − 2 arctan 3
]Ex. 519.
∫ 3
−3sin3 x cos x dx.
[0]
Ex. 520.∫ 2π
0sin3 cos 2x dx.
[0]
Ex. 521.∫ π/2
π/4
xsin2 x
dx.[π
4+ log
√
2]
Ex. 522.∫ π/2
−π/2x sin x cos x dx.
[π4
]Ex. 523.
∫ π
0x sin2 x dx.
[π2
4
]Ex. 524.
∫ e
1e
x| log x|dx.[e2
4+
12−
34e2
]
Ex. 525.∫ 5
2
e2x√
ex − 1dx.
[23
((2 + e5)
√
e5 − 1 − (2 + e2)√
e2 − 1) ]
Ex. 526. Calculate∫ 10
−10f (x) dx, where:
f (x) =
x2 + 2 , x ≤ −2,√
x2 − 4x
, −2 < x < 2,√
x , x ≥ 2.
Ex. 527. Calculate∫ 10
−10f (x) dx,where:
f (x) =
1
√
x2 + 4, x ≤ 0,
x2
x2 + 1, x > 0.
39
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 528. Calcolate∫ 5
−3f (x) dx,where:
f (x) =
sin
x2, x > 0,
cosx2, x < 0.
Ex. 529. Calcolate the area of the surface between the graph of the curves of equation
y = x3, y = 2 − x2 under the condition x < 0. Say if it exists (finite) the area of the surface
between the graph of the two curves, without the condition x < 0.
Ex. 530. Calcolate the area of the surface between the graph of the curves of equation
y = −x2 + x + 2 ed y = x2− 1.
40
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
5.6 Improper Integrals
5.6.1 Determine the convergence or divergence for the following improper integrals
Ex. 531.∫ 1
−1
1√
1 − x2dx. Calculate, if it exist, the value of the integral.
[π]
Ex. 532.∫ log 3
0
1ex − 3
dx.[Divergent
]Ex. 533. I =
∫∞
2
1x logα x
dx, α ∈ R.[If α > 1 , I =
1
(α − 1) logα−1 2; if α ≤ 1 , then I is divergent.
]Ex. 534.
∫ 6
4
1(x − 4) − log(x − 3)
dx.[Divergent
]Ex. 535.
∫ 4
2
1
|cosπx/2|3/5dx.
[Convergent
]Ex. 536.
∫ +∞
1
1((log x)(x5 + x − 2))1/5
dx.[Divergent
]Ex. 537.
∫ +∞
0
sin x log x(x + 1)3/2 − 1
dx.[Convergent
]Ex. 538.
∫ +∞
1
e1/x2− e1/x√
xdx.
[Convergent
]Ex. 539.
∫ π/2
0
e−1/x√
sin xdx.
[Convergent
]Ex. 540.
∫ +∞
0
1mx + ex dx, m ∈ R+.
[Convergent
]Ex. 541.
∫ +∞
2
1√(log x)2(x3 + x)
dx.[Convergent
]5.6.2 Discuss the integrability in an improper sense of the following integrals.
Ex. 542.∫ +∞
1
log(t + 1)t3 + 2t + 1
dt.[Convergent
]Ex. 543.
∫ 1
0
log t(1 − t)5/4t1/2
dt.[Convergent
]Ex. 544.
∫ +∞
0
1√
t(t2 + 1) log(1 +√
t)dt.
[Divergent
]
41
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 545.∫ +∞
0
sin1√
y
(y − 1)1/2dy.
[Divergent
]Ex. 546.
∫ +∞
1
log(2 + x2)√
x arctan x2dx.
[Divergent
]Ex. 547.
∫ +∞
0
e−y2/2√2y + arctan(y1/4)
dy.[Convergent
]Ex. 548.
∫ +∞
1/2
e−x
(x − 3)1/3(x − 1/2)1/2dx.
[Convergent
]Ex. 549.
∫−1
+∞
e−x
(x − 4)2(x + 1/2)1/3dx.
[Divergent
]Ex. 550.
∫ +∞
1/2
1(y − 3)1/3(y − 1/2)1/2
dy.[Divergent
]Ex. 551.
∫ +∞
1/2
1|x − 3|3/4(x − 1/2)1/2
dx.[Convergent
]Ex. 552.
∫ +∞
3
log(3 + x−1/4)(x − 3)3/4(x − 1/2)1/2
dx.[Convergent
]Ex. 553.
∫ 1
0
log x2
(1 − x)9/4x1/2dx.
[Divergent
]Ex. 554. If Ia =
∫ +∞
a
e−x
(x − 3)2(x − 1/2)1/2dx, find a ∈ R such that Ia < +∞.
[a > 3
]Ex. 555. If Ia =
∫ +∞
1
dy(1 + y)2(y + 2)a dy, find a ∈ R such that Ia < +∞.
Moreover, calculate I1.[a > −1. I1 =
12
+ log23
]5.6.3 Determine the values of α ∈ R s.t. the following improper integrals result to be
convergent.
Ex. 556.∫ 1
0
(tan x)α
log(1 + sin x)dx.
[α > 0
]Ex. 557.
∫ +∞
0
arctan(1/xα)√
x + 2dx.
[α >
12
]Ex. 558.
∫ 1
0
cos x + 3xα +
√x
dx.[α < 1
]Ex. 559.
∫ +∞
2
arctan(x + 7)x logα(x − 2)
dx.[α > 1
]
42
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 560.∫ +∞
2
logα(1 + 1/x)√
x + 1dx.
[α >
12
]Ex. 561.
∫ +∞
1
|sin(1/x) − 1/x|α2/2
3√xdx.
[|α| >
23
]Ex. 562.
∫ +∞
1
(1 − cos
1x3
)αxα/2 dx.
[α >
211
]Ex. 563.
∫ +∞
0(arctan x)α(
√x + 3)2α dx. [Divergent for any α]
Ex. 564.∫ +∞
0(e−x +
x2α + 1√
x) dx. [Divergent for any α]
Ex. 565.∫ +∞
−1
arctan(x2 + 3)(x + 1)α(x + 2)
dx.[0 < α < 1
]Ex. 566.
∫ +∞
0arctan(1/x)α(x2 + 3)2α dx.
[−
14< α < 0
]Ex. 567.
∫ +∞
3
e−t
(t − 3)α√
tdt.
[α < 1
]Ex. 568.
∫ +∞
0
sinα(1/√
t)√
t logα(t + 1)dt. [Divergent for any α]
Ex. 569.∫ 2
−1
(ex+3 + 7 sin2 x)xα(ex + 1)
dx.[α < 1
]Ex. 570.
∫ +∞
−∞
e−αx2/2 dx.[α > 0
]Ex. 571.
∫ +∞
1(e1/x
− 1)αlog(2 + x)
x2 dx.[α > −1
]Ex. 572.
∫ +∞
4
logα+1(x − 3)√
ex−4 − 1dx. Moreover, calculate its value for α = −1.
[α > −
32. π
]
Ex. 573.∫ +∞
0
sin( xx2 + 1
)(x2 − sin x2)α
dx.[0 < α <
13
]Ex. 574.
∫ +∞
0
3 + 2 sin x(x − 1)1/3(x + 2)4α
dx.[α >
16
]Ex. 575.
∫ +∞
0
log(1 + xα)x3 dx.
[α > 2
]Ex. 576.
∫ 1
0
1x(− log x)α + x2(1 − x2)1/3
dx.[α > 1
]
43
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
5.6.4 Determine the values of α and β such that the following improper integrals
converge
Ex. 577.∫ 1
0
| log x|α
| sinπx|βdx.
[β < 1, β − α < 1
]Ex. 578.
∫ +∞
0
eαx+β/x
x + 1dx.
[α < 0, β ≤ 0
]Ex. 579.
∫ +∞
0
(arctan x)α
xβ(2 + cos x)dx.
[β > 1, β − α < 1
]5.7 Numerical Series
5.7.1 Determine the nature of the following numerical series
Ex. 580.∞∑
k=1
1
k +√
k.
[Divergent
]
Ex. 581.∞∑
k=1
kk + log k
.[Divergent
]
Ex. 582.∞∑
k=1
1klog k
.[Convergent
]
Ex. 583.∞∑
k=1
(log(log k)
log k
)k
.[Convergent
]
Ex. 584.∞∑
k=1
(k!)2
(2k)!.
[Convergent
]
Ex. 585.∞∑
k=1
k2e−√
k.[Convergent
]
Ex. 586.∞∑
k=1
(√k2 + 1 − k
)log
(1 +
1k
).
[Convergent
]
Ex. 587.∞∑
k=1
(√k + 1 −
√
k)2.
[Divergent
]
Ex. 588.∞∑
k=1
√
1 + sin3k− 1
(1 − e−1/k).
[Convergent
]
Ex. 589.∞∑
k=1
(e1/k2− 2 cos
1k
+ 1).[Convergent
]
44
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 590.∞∑
k=1
13 + eαk
, α ∈ R.[Convergent if α > 0, divergent otherwise
]
Ex. 591.∞∑
k=1
k2
4 + eαk, ∈ R.
[Convergent if α > 0, divergent otherwise
]
Ex. 592.∞∑
k=1
1 −√
e(cos
1k
)k2 . [Convergent
]
Ex. 593.∞∑
k=1
( 59 − 2 cos k
)k.
[Convergent
]
Ex. 594.∞∑
k=1
(log(1 +
33√
k2) −
α3√
k2
).
[Convergent if α = 3, divergent otherwise
]
Ex. 595.∞∑
k=1
(k sin
(1k
))k3
.[Convergent
]5.7.2 Determine the nature of the following series
Ex. 596.∞∑
k=1
(1 −
1k1/3
)k2
.[Convergent
]
Ex. 597.∞∑
k=1
( 35 + cos2 k
)k.
[Convergent
]
Ex. 598.∞∑
k=1
(63n +
(−1)n+1
4n
). Moreover, if it exists, calculate the sum. [Converges to
233
]
Ex. 599.∞∑
n=1
n√
n
en2 .[Convergent
]
Ex. 600.∞∑
k=1
(3x2− 3
x2 + 1
)2n
+n + 1
n2(log n)x + 2
, x ∈ R.[Converges if1 < x <
√2, diverges otherwise
]5.7.3 Determine the nature of the following series for any α ∈ R and x ∈ R
Ex. 601.∞∑
k=4
1k2
(1 −
1k
)kα
.[Converges for any α ∈ R
]
Ex. 602.∞∑
n=1
nα(x + 1)2n
(2n)!, x ∈ R.
[Converges for any α, x ∈ R
]
45
A. Berretti, F. Ciolli Exercises inMathematical Analysis I
Ex. 603.∞∑
n=1
n(1 −
(1 +
1n2α
)1/4).
[Converges if α > 1, divergent otherwise
]
Ex. 604.∞∑
n=1
n8
(n − log n)10 − nα.
[Converges if α , 10, divergent otherwise
]
Ex. 605.∞∑
n=1
nα((
n4− 5n2
)1/4−
(n3− 3n
)1/3).
[Convergent if α < 0, divergent otherwise
]Ex. 606. Determine the values of α ∈ R such that the following two series have the same
nature:∞∑
n=1
(e(nα+1/n)
− 1),
∞∑n=1
log(1 + nα).[α ≥ −1
]
Ex. 607. Study the nature of the series∞∑
n=1
3(−1)n3αn for any α ∈ R. Moreover, calculate its
sum once calculated the one of the two series∞∑
n=1
3 · 32αn and∞∑
n=1
13· 3(2n+1)α.[
Converges for α < 0 to the value9 + 3α
3(1 − 32α)
]5.7.4 Discuss the simply and absolute convergence of the following series
Ex. 608.∞∑
n=0
arctan1
n + 1.
[Simply and absolutely convergent
]
Ex. 609.∞∑
n=0
(α
2α + 3
)n 1log n
.[Simply convergent if α < −3, absolutely convergent if α < −3, α ≥ −1
]Ex. 610.
∞∑k=1
(−1)k(e1/k1/4
− 1)α.
[Simply convergent if α > 0, absolutely convergent if α > 4
]
46