taller calculo ii
TRANSCRIPT
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7/22/2019 Taller Calculo II
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1
1
x3
dx
11
2x2 3x3 dx
10
2
2
3 x 233
dx
21
x+ 1
x x+ 1 dx
10
dxx2 + 7
4
3
dx
x2 4
10
dx8 x2
ln 31
3xex dx
0
sen x dx
0
cos x dx
4
0
tan x dx
11
(nx)1n
n dx n N
21
(xm xn)3x
dx n, m N
21
(xm xn)2
x
dx n, m N
2c bm
c bm
x
mx+b dx; m = 0, c > 0
2c bm
c b
m
ax+d
mx+b dx; m = 0, c > 0
10
x
(x+ 1)2
dx
1
20
b1 x dx
a
b0
a bx dx; a,b >0
e1
x+ ln x
x dx
10
x2
x2 + 2 dx
1
20
x3
2 x2 dx
10
x
2x2 + 3 dx
a
20
xa4 x4 dx; a > 0
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3
2
x2x6
1dx
1
0
arctanx2
4 +x2 dx
x arctan2x
1 + 4x2 dx
aemx dx
ex ex dx
x7x
2
dx
e
1
x
x2 dx
ex
ex 1 dx ex
a bex dx dx
2x + 3
ax
1 +a2x dx
sen(a+bx)dx
sen log x
x dx
sen2 x dx
xdx
cos2 x2
tanx dx
tan
x
x dx
sen3 6x cos6x dx
sen3x
3 + cos 3x dx
tan x
cos2 x dx
1 + sen 3x
cos2 3x dx
(2senh5x 3cosh5x) dx
dx
senh x cosh x
x
3
1x4 4x+ 1 dx x3
x8 + 5 dx
dx
ex
1 sen x
x+ cos x dx
dx
x ln2 x
x dx
1 x4 x2
x2
2
dx
dxex + 1
e2x
1 +ex dx
sen3 x
cos x dx
dx
(a+b)(a b)x2 0< b < a sen x cos x
cos2 x sen2 x dx
1 + 3 cos
2
x sen2x dx
dx
(1 +x2) ln
x+
1 +x2
earctan x +x ln(1 +x2) + 1
1 +x2 dx
x2
1 x2 dx
x32 x2 dx
dx
x
x2 1
dx
x2
4 x2
x2
a2
x dx
x2 + 1
x dx
1 x2 dx
25
x2
x dx
dx
x2 1
dx
x2 2x+ 5
x2
+ 4 dx
x
1 +x4 dx
11
x2
25 x2 dx 1
0
x2
(4 x) 32dx
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1
1x3
16 x2
dx
1
0
dx
x
25 4x
2
arctan x dx
x sen x dx
x dx
ex
x2e3x dx
x
3
ex
3 dx
ln2 x dx
ln x
x dx
x arc senx dx
x
sen2 x dx
ex sen x dx
eax sen bx dx
cos
x dx
xex dx
x sec x tan x dx
x
2
ln x dx
arctan x dx
(ex + 2x)
2dx
x sen x cos x dx
x2
+ 5x+ 6
cos2x dx
e1
x2 ln x dx
0
sen(3x)cos(5x)dx
1
11
12x2 + 3653x+ 2 dx
e1
ln 3x dx
0
(2x 5)sen x dx
10
(2x+ 4)e2x+4 dx
1
1
arctan x dx
1
20
arcsen x dx
n, m N
senn x dx= senn1 x cos x+ (n 1)
senn2 x cos2 x dx
senn x dx= sen
n1 x cos x
n +
n 1n
senn2 x dx
cosn x dx= cos
n1
x sen xn
+ n 1n
cosn2 x dx
senn+1 x
cosm+1 x dx=
1
m
senn x
cosm x n
m
senn1 x
cosm1 x dx
cosm+1 x
senn+1 x dx=
1
n
cosm x
senn x m
n
cosm1 x
senn1 x dx
xn ln x dx
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7/22/2019 Taller Calculo II
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2
1
x3ex2
dx
21
ln2 x
x2 dx
10
sen2 x
ex dx
2
1
x2
(x2 + 1)2 dx
21
x29 x2 dx
40
x2 arctan 3x dx
2
0
(arc senx)2 dx
a
20
a2 x2 dx
4
8
arcsen
x1 x dx
2
1x
2 + 2x+ 3 ln x dx
10
x(arctan x)2 dx
a0
dx
(a2 +x2)2
b
h
A= 12
bh
r >0 A= r2
B
b
h
A= 12
(B+b)h
r V = 43
r3
r h V = 1
3r2h
(x, y)
x2 y2 36 0, y2 9x.
a >0 y = a sen x
y= cos x, 0 x 2 .
x2
9 +
y2
4 = 1,
x2
4 +
y2
9 = 1.
y = x2 x = y2 x= 1
y = 6 3x x2 x+y 3 = 0
x= 3
y= 0
x2 +y2 = 9 x= 4
h 2a 2b
x x 2x2 x= 0 x= a
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y= 3 2x x2 (2,5) y
y2
=x(x 1)2
2
6 x= y2
y = cos x x y= sen x x=
2
a2y2 =ax3 x4 x (a > 0) x y= a cosh xa (a > 0)
y = sen x(0 x 2
) y= 1 y
x= 4 y2
y [2, 2] x
2a a a > 0 h
y
x = y2 x = 8 y2
a > 0
y = sen x
y= 0
x= 0
x= 1/4
x
y
y2 = 4ax a >0 x= a
x2
4 +y2 = 1 x= 3
f(x) = 3x3 x2 10x g(x) = 2x x2
x
y2 = x
y2 = 2x x2 x= 3
f(x) = 1 x/2 g(x) = 1 +x/2 x= 0 x x= 2
y= x3 +x+ 1 y= 1 x= 1
f(x) =x3
6 +
1
2x
1
2, 2
a > 0 b > 0
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y= x3 y= x2
y= x+ 6
y =
log x
y= x2 +x+ 3 y= x3 + 3x2 2x+ 3 (1, e) (e, 1) y= ex y= log x y= 1 x
x2 +y2 36 y2 9x
r h
r h
4
y = 2
y = x2
+ 2x
x2 +y2 +z2 1 x2 +y2 r2 r (0, 1)
[0, 4] f(x) = 42
3 x3/2 1
y= 3x (1, 3) (2, 6)
9y2 = 4x3
8y = x4 + 2x2 x = 1 x= 2
y = 13
(x2 + 2)3/2 x = 0
x= 3
x2/3 +y2/3 = 1 x= 1/8 x= 1
10
dxx
21
dx
x
10
dx
xp
30
dx
(x 1)2
10
dx1 x2
1
20
dx
x ln x
2
0
cot x
10
dx
x3 5x2
0
dx
x
0
dx
xp
0
sen x
a
dx
x ln x (a > 1)
a
dx
x ln2 x (a > 1)
0
ekx dx
0
arctan x
x2 + 1 dx
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7/22/2019 Taller Calculo II
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2
dx
(x2
1)2
0
dx
x3 + 1
dx
1 +x2
dx
x2 + 4x+ 9
c
2
cx
x2 + 1 1
2x+ 1
dx
c
c
1
x
2x2 + 2c c
x+ 1
dx
c
c
0
11 + 2x2
cx+ 1
dx
c
1
(x+a)(x+b)
x2 5x+ 9x2 5x+ 5
1
(x 1)(x+ 2)(x+ 3)
2x2 + 41x 91(x 1)(x+ 2)(x 4)
5x3
+ 2x3 5x2 + 4x
1
x(x+ 1)2
x3 114x3 x
x2 8x+ 7(x2 3x 10)2
x3 +x+ 1
x(x2 + 1)
x4
x4 1
1
(x2 4x+ 3)(x2 + 4x+ 5)
1x3 + 1
x3 + 1
(x2 4x+ 5)2
1
(x+ 1)(x2 +x+ 1)2
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7/22/2019 Taller Calculo II
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cos3 x dx
sen5 x dx
sen2 x cos3 x dx
sen3
x
2cos5
x
2 dx
cos5 x
sen3 x dx
sen4 x dx
1
sen4
x
dx
1
sen2x cos4 x dx
sec5 4x dx
tan2 5x dx
cot3 x dx
cot4 x dx
tan3
x
3+ tan4
x
4
dx
x sen2 x2 dx
sen5 x 3
cos x dx
sen3x cos5x dx
sen10x sen15x dx
cos
x
2cos
x
3 dx
sen
x
3cos
2x
3dx
cos(ax+b) cos(ax b)dx
sen ax sen(ax+b)dx
an= 1
log(n+k) bn=
1
log n k R+
an= 1
log(kn) bn=
1
log n k R+ k = 1
n
p
k
an= p(n) bn= p(n+k)
an= apnp + ap1n
p1 + ap2np2 + + a1n + a0 bn= apnp ap= 0
an= 1
(n+ 1) n bn= 1
n1
= 0
n=1
n
7n+ 4
4n2
n=2
1
(ln n)lnn
n=2
ln
n+ 1
n 1
n
n=1
n
(n+ 1)3
n=1
n+ 2(n+ 1)
n+ 3
n=1
4n2 n+ 3n3 + 2n
n=1
2n1
(3n+ 2) n4/3
n=1
3n
n5n
n=1
n+
n
2n3 1n=2
1
ln n
n=1
ln na
n!
n=1
ln n
2n3 1n=1
(n!)24n
(2n)!
n=1
lnn+ 1
n+ 2
n=1
an
n a > 0 a = 1
n=1
n!an
nn a >0 a = 1
n=1
na
n!
n=1
sen4 n
n2
n=1
nj=1
(a+j)
n!
n=1
nan
a > 0 a
= 1
n=1
(ln n)n
n=1
n+ 1 n
1
2 3+ 2
3 4+ 2
5+
22 + 1
2 52 +32 + 1
3 53
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1 + 2!
22+
3!
33+
4!
44
1
2+
2
3+
3
4
n=1
nn
n!
n=1
n2en
n=2
1
(n log n)p
n=1
1
(n+ 1)(n+ 2)
n=1
1
2n(n+ 1)
n=0
12
100n
n=0
3n + 4n
5n
n=2
ln
n+ 1
n 1
n=0
n
2n1+
3
2n
n=1
ln(n+ 1)2
n(n+ 2)
n=0
1 2n3n
n=1
n
2n 1+ 3
2n
n=0
2n +n(n+ 1)
2n+1n(n+ 1)
n=0
ln
1 +
1
n
n
(n+ 1)
[ln(n+ 1)n+1] ln nn
n=1
1
n 1
n!
n=1
n3
2n
n=1
(1)n ln nn
n=1
n!
(2)n
n=1
(1)nn+
n+ 1
n=1
sen
4n2
n=1
(1)nn(n+ 1)
n=1
(1)n n2n
n=1
(1)n 1n
1 13
+1
5 1
7+
1
2 l n 2 1
3 l n 3+
1
4 l n 4 1
ln 5+
1
4 1
6+
1
8 1
10+
n=1
1
n2nxn
n=1
1
nxn
n=1
(1)nn
x2n
n=1
1
n2xn
n=1
(n!)2
(2n)!x2n
n=0
n2 + 1
(n+ 1)!xn
n=0
2n
n+ 1xn
n=2
nxn
n=2
1
n2n(x 3)n
n=2
2n
n+ 1
x+
1
2
n
n=1
xn ln n
n=1
n2
2nxn
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n=0
2n 13n+ 1
xn
n=1
nn
n!
x+
1
2
n
n=2
2 5 8 11 (3n 1)3 7 11 15 (4n 1) x
n
n=0
(1)nn!n2
(x 1)n
n=0
(1)nn+ 1
(x 1)n
n=0
(1)nn33n
(x+ 3)n
n=0
xnn
ln
2n+ 1
n
x
f(x) = ex
f(x) = sen x
f(x) = ln(1 x) f(x) =
1
1 x
f(x) = 1
1 +x2
f(x) = arctan x
x
f(x) = 1
1 + sen x f(x) =
cos x
1 + sen x f(x) =
cos x
1 2x