indefinite integration.pdf
TRANSCRIPT
-
8/21/2019 Indefinite integration.pdf
1/12
INDEFINITE INTG. # 1
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (1)
D EF IN I T IONS ND RESULTS
1. If f & g are functions of x such that g(x) = f(x) then ,
f(x)dx = g(x)+ c ddx
{g(x)+c} = f(x), where c is called the constant of integration.
2. STANDARD RESULTS :
(i) (ax + b)ndx = ax b
a n
n
1
1+ c n 1 (ii) dx
ax b=
1
aln (ax + b) + c
(iii) eax+bdx = 1a eax+b + c (iv) apx+q dx = 1p
a
na
px q
(a > 0) + c
(v) sin (ax+ b) dx = 1a
cos (ax+ b)+ c (vi) cos (ax + b) dx = 1a
sin (ax+ b) + c
(vii) tan(ax+ b) dx = 1a
ln sec(ax + b)+ c (viii) cot(ax+b)dx = 1a
ln sin(ax +b)+ c
(ix) sec (ax + b) dx = 1a
tan(ax + b) + c
(x) cosec(ax + b) dx = 1a
cot(ax + b)+ c
(xi) sec (ax + b) . tan (ax + b) dx = 1a
sec (ax + b) + c
(xii) cosec (ax + b) . cot (ax + b) dx = 1a
cosec (ax + b) + c
(xiii) secx dx = ln (secx + tanx) + c OR ln tan 4 2
x
+ c
(xiv) cosec x dx = ln (cosecx cotx) + c OR ln tan x2
+ c OR ln (cosecx + cotx)
(xv) d xa x2 2 = sin1 x
a+ c (xvi) d xa x2 2 =
1a
tan1 xa
+ c
(xvii) d xx x a
2 2=
1
asec1
x
a+ c
(xviii) d xx a2 2
= ln x x a 2 2 OR sinh1 xa
+ c
(xix) d xx a2 2
= ln x x a 2 2 OR cosh1 xa
+ c
IIT ians P A C E216 - 217, 2nd floor, Shoppers point, S. V. Road. Andheri (West) Mumbai 400 058 . Tel: 2624 5223 / 09
-
8/21/2019 Indefinite integration.pdf
2/12
INDEFINITE INTG. # 2
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (2)
(xx) d xa x2 2
=1
2aln
a x
a x
+ c (xxi) d xx a2 2
=1
2aln
x a
x a
+ c
(xxii)
a x2 2
dx =
x
2
a x
2 2
+
a 2
2sin1
x
a+ c
(xxiii) x a2 2 dx = x2
x a2 2 +a2
2sinh1
x
a+ c
(xxiv) x a2 2 dx = x2
x a2 2 a 2
2cosh1
x
a+ c
(xxv) eax. sin bx dx = ea b
ax
2 2(a sin bx b cos bx) + c
(xxvi) eax . cos bx dx = ea b
ax
2 2
(a cos bx + b sin bx) + c
3. INTEGRALS OF THE TYPE :
(i) [ f(x)]nf(x) dx OR
f xf x
n
( )
( ) dx put f(x) = t & proceed .
(ii)dx
ax bx c2 ,
dx
ax bx c2 , ax bx c2 dx
Express ax2+ bx + c in the form of perfect square & then apply the standard results .
(iii)px q
ax bx c
2 dx ,px q
ax bx c
2
dx .
Express px + q = A (differential coefficient of denominator) + B .
(iv) ex [f(x) + f(x)] dx = ex . f(x) + c (v) [f(x) + xf(x)] dx = x f(x) + c
(vi) d xx xn( )1
n N Take xn common & put 1 + x n= t .
(vii)
dx
x xnn
n21
1 ( ) n N , take xn common & put 1+xn= tn
(viii)
dx
x xn n
n
1
1
/ take xncommon as x and put 1 + xn= t .
(ix) d x
a b x sin2 OR
d x
a b x cos2OR
d x
a x b x x c xsin sin cos cos2 2
Multiply Nr. . & D
r. . by sec x & put tan x = t .
(x) d x
a b x sinOR
d x
a b x cos OR
d x
a b x c x sin cos Hint :
Convert sines & cosines into their respective tangents of half the angles , put tanx
2= t
-
8/21/2019 Indefinite integration.pdf
3/12
INDEFINITE INTG. # 3
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (3)
(xi) a x b x cx m x n
.cos .sin
.cos .sin
dx . Express Nr A(Dr) + Bd
d x(Dr) + c & proceed .
(xii)
x
x K x
2
4 2
1
1
dxOR
x
x K x
2
4 2
1
1
dx where K is any constant .Divide Nr & Dr by x & proceed .
(xiii)dx
ax b px q( ) & dx
ax bx c px q2 ; put px + q = t2.
(xiv) dx
ax b px qx r ( )
2
, put ax + b =1
t ;
dx
ax bx c px qx r 2 2
, put x =1
t
(xv)x
x
dx or x x ; put x = cos2 + sin2
x
x
dx or x x ; put x = sec2 tan2
dx
x x
; put x = t2 or x = t2.
EXERCISE I
1. If f(x) =2 2
3
sin sinx x
x
dx where x 0 then Limitx 0 f(x) has the value ;
(A) 0 (B) 1 (C) 2 (D) not defined
2. If 12
sinx
dx = A sin x
4 4
then value of A is :
(A) 2 2 (B) 2 (C)1
2(D) 4 2
3. If y =
dx
x1 23 2
/ and y = 0 when x = 0, then value of y when x = 1 is :
(A)2
3(B) 2 (C) 3 2 (D)
1
2
4. Ifcos
cot tan
4 1x
x x
dx = A cos 4x + B where A & B are constants, then :
(A) A = 1/4 & B may have any value (B) A = 1/8 & B may have any value(C) A = 1/2 & B = 1/4 (D) none of these
5. cot x sec 4 x d x =
(A) 2 tan x +2
5 tan5 x + c (B) 2 tan x +
2
5 tan5 x + c
(C) tan x +2
5 tan5 x + c (D) tan x +1
5 tan5 x + c
-
8/21/2019 Indefinite integration.pdf
4/12
INDEFINITE INTG. # 4
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (4)
6. Given (a > 0) ,1
x xalog dx = loge a loge (loge x) is true for :
(A) x > 1 (B) x > e (C) all x R (D) no real x .
7. cot1 e
e
x
xdx is equal to :
(A)1
2ln (e2x+ 1)
cot1 ee
x
x+ x + c (B)
1
2ln (e 2x+ 1) +
cot1 ee
x
x+ x + c
(C)1
2ln (e 2x+ 1)
cot1 ee
x
xx + c (D)
1
2ln (e2x+ 1) +
cot1 ee
x
xx + c
8.tan cot
tan cot
1 1
1 1
x x
x x
dx is equal to :
(A)4
x tan1 x +
2
ln (1 + x2) x + c (B)
4
x tan1 x
2
ln (1 + x2) + x + c
(C)4
x tan1 x +
2
ln (1 + x2) + x + c (D)
4
x tan1 x
2
ln (1 + x2) x + c
9. If
x
x x
4
22
1
1
dx = A ln x+
B
x1 2+ c , where c is the constant of integration then
(A) A = 1 ; B = 1 (B) A = 1 ; B = 1 (C) A = 1 ; B = 1 (D) A = 1 ; B = 1
10.
n x
x n x
| |
| |1dx equals :
(A)2
31 n x (lnx 2) + c (B)
2
31 n x (lnx+ 2) + c
(C)1
31 n x (lnx 2) + c (D) 2 1 n x (3 lnx 2) + c
11. Antiderivative ofsin
sin
2
21
x
x w.r.t. x is :
(A) x 22arctan 2 tanx + c (B) x 12 arctan
tan x2
+ c
(C) x 2 arctan 2 tanx + c (D) x 2 arctantan x
2
+ c
12. sin x . cos x . cos 2x . cos 4x . cos 8x . cos 16 x dx equals :
(A)sin 16
1024
x+ c (B)
cos 32
1024
x+ c (C)
cos 32
1096
x+ c (D)
cos 32
1096
x+ c
-
8/21/2019 Indefinite integration.pdf
5/12
INDEFINITE INTG. # 5
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (5)
13. 1
1
x
xdx =
(A) x 1 x 2 1 x + cos1 x + c (B) x 1 x + 2 1 x + cos1 x + c
(C) x 1 x 2 1 x cos 1 x + c (D) x 1 x + 2 1 x cos 1 x + c
14.3 5
4 5
e e
e e
x x
x x
dx = Ax + B ln 4 e2x5+ c then :
(A) A = 1, B = 7/8; C = const. of integration(B) A = 1, B = 7/8; C = const. of integration(C) A = 1/8, B = 7/8 ; C = const. of integration(D) A = 1, B = 7/8; C = const. of integration
15.x
x x
1
1
12
. dx equals :
(A) sin11
x+
x
x
2 1(B)
x
x
2 1+ cos1
1
x+ c
(C) sec1 x xx
2 1+ c (D) tan1 x2 1
x
x
2 1+ c
16. tan 32x sec 2x dx =
(A)1
3sec32
x
1
2sec 2
x + c (B)
1
6sec32
x
1
2sec 2
x + c
(C) 16 sec32x 12 sec 2x + c (D) 13 sec
32x + 12 sec 2x + c
17. ( )x
x xex
1
12
dx =
(A) ln
x e
x e
x
x1
+
1
1 ex+ c (B) l
n
x e
e
x
x1
+
1
1 x ex+ c
(C) ln
x e
x e
x
x1
+
x
x ex1+ c (D) l
n
x e
x e
x
x1
+
1
1 x ex+ c
18.dx
x xcos . sin3 2 equals :
(A)2
5(tan x)5/2+ 2 tanx + c (B)
2
5(tan2 x + 5) tanx + c
(C)2
5(tan2 x + 5) 2tanx + c (D) none
-
8/21/2019 Indefinite integration.pdf
6/12
INDEFINITE INTG. # 6
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (6)
19. Ifdx
x xsin cos3 5 = a cot x + b tan3 x + c where c is an arbitrary constant of
integration then the values of a and b are respectively :
(A) 2 & 23 (B) 2 & 23
(C) 2 & 23
(D) none
20. cos
sin sin
3
2
x
x x d
x =
(A) ln sin x+ sin x + c (B) l
n sin xsin x + c
(C) ln sin xsin x + c (D) l
n sin x+ sin x + c
21. cos cos
sin sin
3 5
2 4
x x
x x
dx :
(A) sinx 6 tan1 (sin x) + c (B) sin x 2 sin1 x + c(C) sinx 2 (sinx)16 tan1 (sin x) + c (D) sinx 2 (sinx)1+ 5 tan1 (sin x) + c
22. 1
6 6cos sinx x d
x =
(A) tan1(tan x + cot x) + c (B) tan1(tan x + cot x) + c(C) tan1(tan x cot x) + c (D) tan1(tan x cot x) + c
23. Primitive of
3 1
1
4
42
x
x x
w.r.t. x is :
(A)x
x x4 1
+ c (B) x
x x4 1
+ c (C)x
x x
1
14+ c (D)
x
x x
1
14+ c
24.dx
x5 4 cos = tan1 m
xtan
2
+ C then :
(A) = 2/3 (B) m = 1/3 (C) = 1/3 (D) m = 2/3
25.x x
x
2 2
21
cos
cosec2 x dx is equal to :
(A) cot x cot 1 x + c (B) c cot x + cot 1 x
(C) tan 1 x cos
sec
ec x
x+ c (D) e n x tan
1cot x + c
where 'c' is constant of integration .
26.dx
x x 2 equals :
(A) 2 sin 1 x + c (B) sin1 (2x 1) + c
(C) c 2 cos1(2x 1) (D) cos12 x x 2 + c
-
8/21/2019 Indefinite integration.pdf
7/12
INDEFINITE INTG. # 7
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (7)
27. 2mx. 3nx dx when m, n N is equal to :
(A)2 3
2 3
mx nx
m n n n
+ c (B)
e
m n n n
m n n n x
2 3
2 3
+ c
(C)
2 3
2 3
mx nx
m nn
.
.+ c (D)
mnm n n n
x x. .2 3
2 3 + c
28. If eu. sin 2x dx can be found in terms of known functions of x then u can be :(A) x (B) sin x (C) cos x (D) cos 2x
29. sec2 24
x
dx equals :
(A) c 1
2cot 2
4x
(B)1
2tan 2
4x
+ c (C)1
2(tan 4x sec 4x) + c
(D) none
30. n x
x x
(tan )
sin cos dx equal :
(A)1
2ln2(cot x) + c (B)
1
2ln2(sec x) + c
(C)1
2ln2(sinx secx) + c (D)
1
2ln2(cos x cosec x) + c
EXERCISE II
LEVEL I
Evaluate the following :
1. cos cos
cos cos
2 2x
x
dx 2. sin cos
sin cos
6 6
2 2
x x
x x
dx
3. x
a bx
2
2( )dx 4. sin 4x cos4x dx
5.
cos 2x cos 4x cos 6x dx
6.
1
sin ( ) cos ( )x a x b dx
7. tan
tan
x
a b x 2dx 8. 1 2
1 2 tan (tan sec ) /x x x dx
9. 1
3cos cosx xdx 10.
e x
x
sin
1 2
21dx
11. x x
x
1
2dx 12. 5
1x x tan .x
x
2
2
2
1
dx
-
8/21/2019 Indefinite integration.pdf
8/12
INDEFINITE INTG. # 8
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (8)
13. x
a x3 3dx 14. cosec x 1 dx
15. 2 3
3 182x
x x
dx 16. sin
sin
x
x3 dx 17. x x
x
sin
cos1 dx
18. x x
x
2 1
23 2
1
sin/
dx 19. ex
x
x( ) 1 2dx 20.
( )x
x x
1
1
2
4 2dx
LEVEL II
Evaluate the following :
21. 1
3sin sin ( )x x dx , n
, n Z
22.
1
12 43 4
x x /
dx 23. 2 2
6 42sin cos
cos sin
dx
24. x a xa x
2 2
2 2
dx 25. sin 1
x
a x dx
26. tan tan
tan
3
31d 27.
sin
sin
x
x4dx
28. 1 2sin sinx x dx 29. 11 12 2( ) ( )x x dx
30. tan cot d 31. ex
x x
x
3
22
2
1
dx
32. sin cos
sin
x x
x
9 16 2
dx 33. 3 4 23 2
sin cos
sin cos
x x
x xdx
34. cos 2x ln (1 + tan x) dx 35. d xx xsin tan
36. 1
1 4 sin xdx 37. cos
sin
3
11
x
xdx
38. x
x x 3dx 39.
1
cos cos
cos cos
x
xdx
40. d x
x x x( ) ( ) ( ) 41. sec4 x cosec2x dx
42. cos sin
( cos sin )
2
2
2
2
x x
x x
dx 43. cos cos
cos
5 4
1 2 3
x x
xdx
-
8/21/2019 Indefinite integration.pdf
9/12
INDEFINITE INTG. # 9
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (9)
44. cos x . ex. x2dx 45. sin ( )sin ( )
x a
x a
dx
46.
cot
( sin ) (sec )
x dx
x x1 1 47.
cos cot
cos cot
.sec
sec
ec x x
ec x x
x
x
1 2dx
48. d x
x xsin sec 49. tanx. tan 2x. tan 3x dx
50.
dx
x xsin sin 2
51. x
x x x
2
2( sin cos )
dx
52. n x x
x
cos cos
sin
22 dx 53.
sin
sin cos
x
x x dx
54.
e
x x x
x
xsin.
cos sin
cos
3
2
dx 55.
d x
a b x cos2 (a > b)
56. cos
sin
2 x
xdx 57.
cot tan
sin
x x
x
1 3 2 dx
58. 5 4
1
4 5
52
x x
x x
dx 59.
dx
x42
1 60. ex
x
x
2
2
1
1
( )dx
61. x x 2 2 dx 62. x l n x x
x
2 2
4
1 1 2
lndx
63. l n x x(ln ) (ln )
12
dx 64.
d x
x x 1 23 51 4/
65.
dx
x x x x x3 2 23 3 1 2 3 66.
( )
( )
ax b dx
x c x ax b
2
2 2 2 2
67. e x
x x
x 2
1 1
2
2
( )dx 68.
x
x x7 10 23 2
/ dx
69. x x
x
ln/2 3 21
dx 70. 113
x
x
dx
x71.
2 3
2 3
1
1
x
x
x
xdx
72. x
x x
d x
x
2
3 3 12 73.
dx
x x3 31( ) 74. 2
2
2
x x
xdx
75. Integrate1
2f(x) w.r.t. x4 , where f (x) = tan 1x + ln 1x ln 1x
-
8/21/2019 Indefinite integration.pdf
10/12
INDEFINITE INTG. # 10
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (10)
ANSWER SHEET
EXERCISE I
1. B 2. D 3. D 4. B 5. B 6. A 7. C
8. D 9. C 10. A 11. A 12. B 13. A 14. D
15. C 16. C 17. D 18. B 19. A 20. B 21. C
22. C 23. B 24. AB 25. BCD 26. ABD 27. BC
28. ABCD 29. ABC 30. ACD
EXERCISE II
1. 2 sin x + 2x cos + c 2. tan
x cot
x 3
x + c
3.13b
b x a bx aa
a bx
2
2
log( )
| | + c 4. 1128
3 41
88x x x
sin . sin + c
5.1
4 x
x x x
sin sin sin12
12
8
8
4
4+ c 6.
1
cos ( )a blog
e
sin ( )
cos ( )
x a
x b
+ c
7.1
2 ( )b alog a x b xcos sin2 2 + c 8. log sec x + tan x+ log sec x+ c
9.1
4
[cosec x - log sec x + tan x
]+ c 10.
e xsin
1 2
2+ c
11. (x + 1) + 2 x 1 2 log x + 2 2 tan 1
x 1 + c 12. 15log
51x x
tan + c
13.2
3sin 1
x
a
3 2
3 2
/
/
+ c 14. log sin sin sinx x x
1
22 + c
15. log x x2 3 18 2
3log
x
x
3
6+ c 16.
1
2 3log
3
3
tan
tan
x
x+ c
17. x cot x2 + c18. x x
xsin
1
21 12 sin1
2x + 1
2log 1 2 x + c 19. 1 1x . e
x+ c
20.1
3tan 1
x
x
2 1
3
2
3tan 1
2 1
3
2x
+ c 21.
2
sin sin ( )
sin
x
x
+ c
22. 114
1 4
x
/
+ c 23. 2 log sin sin2 4 5 + 7 tan 1(sin 2) + c
24.1
2a2sin1
x
a
2
2
+
1
2 a x4 4 + c 25. (a + x) arc tan
x
a a x + c
-
8/21/2019 Indefinite integration.pdf
11/12
INDEFINITE INTG. # 11
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (11)
26. 1
31 + tan +
1
6log tan2 tan + 1+ 1
3tan1
2 1
3
tan
+ c
27. 1
8 log
1
1
sin
sin
x
x +
1
4 2 log1 2
1 2
sin
sin
x
x + c
28.1
6log 1 cos x+
1
2log 1 + cos x+
2
5log 3 + 2 cos x+ c
29.1
2log x + 1
1
2 1( )x 1
4log x2 1 + c 30. 2 tan1
tan
tan
1
2+ c
31. exx
x
1
12+ c 32.
1
40log
5 4
5 4
(sin cos )
(sin cos )
x x
x x+ c 33. 2 3
21x arc
xc
tan tan
34.1
2 [sin 2x ln (1 + tan x) - x + ln (sin x + cos x)]+ c 35. 12 2 14 22l n x x ctan tan
36.1
2 2tan 1 2 tan x + 1
2tan x + c 37.
2
5cot5/2x
2
9cot 9/2x + c
38. x 6
5x5/6+
3
2x2/32 x + 3 x1/36 x1/6+ 6 l
n (1 + x1/6) + c
39.x cos + sin logcos ( )
cos ( )
1212
x
x+ c 40.
2
.x
xc
41.1
3 tan3 x + 2 tan x cot x + c 42.
1
2 tan x 1
5 x 2
5 log2 cos x sin x+ c
43. (sinsin
)xx2
2+ c 44.
1
2ex x x x x2 21 1 cos ( ) . sin + c
45. cos a . arc cos coscos
x
a
sin a . ln sin sin sinx x a 2 2 + c
46.1
2ln tan
x
2+
1
4sec
x
2+ tan
x
2+ c 47. sin 1
1
2 2
2sec
x
+ c
48.1
2 3
3
3l n
x x
x xarc x x c
sin cos
sin costan (sin cos )
49.
n x n x n x(sec ) (sec ) (sec )1
22
1
33 + c
50. 1
sinln cot cot cot cot cotx x x 2 2 1 + c 51. sin cos
sin cos
x x x
x x xc
52.cos
sin
2x
xx cot x . ln e x xcos cos 2 + c
53. ln(1 + t) 1
4ln(1 + t4) +
1
2 2ln
t t
t t
2
2
2 1
2 1
1
2tan1 t2+ c where t = cotx
-
8/21/2019 Indefinite integration.pdf
12/12
INDEFINITE INTG. # 12
IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (12)
54. esinx (x secx) + c 55.
b x
a b a b x
a
a barc
a b
a b
xsin
( cos )tan . tan
/2 2 2 2 3 2
2
2+ c
56. 12 22 1
2 11
2
22 n
x x
x xn x x
cot cot
cot cotcot cot
+ c 57. tan
1 2 2sinsin cos
xx x
+ c
58. x
x x
1
15 + c 59. 3 tan1 x
x
x4 14
3
16ln
x
x
1
1+ c 60. ex
x
x
1
1+ c
61.1
3 x x 2
3 2
2/
2
221 2
x x / + c 62.
x xx x
2 2
3 2
1 1
92 3 1
1
. ln
63. xln (lnx) x
l n x+ c 64. 4
3
1
2
1 4
x
xc
/
65. x x
x
2
2
2 3
8 1
( )
+1
16 . cos1
2
1x
+ c
66. sin
1
2ax b
cxk 67. ex
1
1
x
x+ c 68.
2 7 20
9 7 10 2
( )x
x xc
69. arc x
l n x
xcsec
2 1
70. nu
u u
uc where u
x
x
| |tan
2
4 2
12
31
13
1 2
3
1
1
71. 83
1
2 5
5 1
5 111 1 2tan sin
t nt
tx x + c where t =
1
1
x
x
72.2
3 3 1
arcx
x
ctan
( )
73. 15 5 2
4 1
2
2
x x
x x
+
15
8
ln1 1
1 1
x
x+ c
74.
2 2
4
4 2 2 2 2 1
3
2 21x x
xn
x x x
x
x sin + c 75. ln(1 x4) + c