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INDEFINITE INTEGRAL JEE ASSIGNMENT
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286 Indefinite IntegralProperties of Integration, fundamental Integration formulae
Basic Level
1.
sec x dx [MP PET 1988,95; Rajasthan PET 1996]
x
(a) log tan c (b) log(sec x tan x ) c (c) log x c (d) log(sec x tan c) c
4 2 4
2.
5 sin x dx [MP PET 1988]
(a) 5 cos x c (b) 5 cos x c (c) 5 sin x c (d) 5 sin x c
(sec x tan x) dx
2
3. [MP PET 1987, 92]
1
(a) 2(sec x tan x ) x c (b) (sec x tan x )3 c (c) sec x (sec x tan x ) c (d) 2(sec x tan x ) c
3
cosec
2
4. x dx is equal to [MP PET 1999]
(a) cot x C (b) cot x C (c) tan 2 x C (d) cot 2 x C
5.
sec x tan x dx [Rajasthan PET 2003]
(a) sec x tan x C (b) sec x C (c) tan x C (d) sec x C
sin x cos x
6.
1 sin 2 x
dx [MP PET 1990]
(a) sin x c (b) cos x c (c) x c (d) x 2 c
(3cosec x 2 sin 3 x ) dx
2
7. [AI CBSE 1981]
2 2 2
(a) 3 cot x cos 3 x c (b) 3 cot x cos 3 x c (c) 3 cot x cos 3 x c (d) None of these
3 3 3
1 cos 2 x
8.
sin 2 x
dx [MP PET 1993; Ranchi BIT 1982]
(a) cot x 2 x c (b) 2 cot x 2 x c (c) 2 cot x x c (d) 2 cot x x c
9. The value of
cot x dx is [Rajasthan PET 1995]
(a) log cos x c (b) log tan x c (c) log sin x c (d) log sec x c
(x 5)
1
10. The value of 2
dx is [Karna
1 1 2
(a) c (b) c (c) c (d) 2(x 5)3 c
x 5 x 5 (x 5)3
, Indefinite Integral 287
x2
11.
x 2
4
dx equals to [Rajasthan PET 2001]
(a) x 2 tan 1 (x / 2) c (b) x 2 tan 1 (x / 2) c (c) x 4 tan 1 (x / 2) c (d) x 4 tan 1 (x / 2) c
x sec x 3 dx
2
12. [MNR 1986; Roorkee
1975]
1
(a) log(sec x 3 tan x 3 ) (b) 3(sec x 3 tan x 3 ) (c) log(sec x 3 tan x 3 ) (d) None of these
3
cos 2 x 1
13.
cos 2 x 1 dx [MP PET 2000]
(a) tan x x c (b) x tan x c (c) x tan x c (d) x cot x c
sin
1
14. (cos x)dx
x x 2 x x 2 x x 2
(a) (b) (c) (d)
2 2 2 2
(sin
1
15. x cos 1 x ) dx [MP PET 1990]
1
(a) x c (b) x (sin 1 x cos 1 x ) c (c) x (cos 1 x sin 1 x ) c (d) x c
2 2
x (tan 1 x cot 1 x ) dx
51
16. [MP PET 1991]
x 52 x 52
(a) (tan 1 x cot 1 x ) c (b) (tan 1 x cot 1 x ) c
52 52
x 52 x 52
(c) c (d) c
104 2 52 2
x
1
17. The value of 4
dx is [Rajasthan PET 1995]
1 1 1 1
(a) c (b) c (c) c (d) c
3x 3 3x 3 4x3 3x 3
a dx
x
18. [Rajasthan PET 2003]
ax
(a) C (b) a x log a C (c) log a c (d) a x C
log a
a da
x
19. [MP PET 1994, 96]
ax ax
(a) C (b) a x log e a C (c) C (d) None of these
log e a x 1
13
x
20. dx = [Kerala (Engg.) 2002]
13 x
(a) C (b) 13 x 1 C (c) 14 x C (d) 14 x 1 C
log 13
e
x log a
21. . e x dx is equal to
(ae ) x ex
(a) (ae) x (b) (c) (d) None of these
log( ae ) 1 log a
,288 Indefinite Integral
a
3 x 3
22. dx [Roorkee 1977]
a 3 x 3 a 3 x 3
(a) c (b) c (c) a 3 x 3 log a c (d) 3 a 3 x 3 log a c
log a 3 log a
e dx
log(sin x)
23. [MP PET 1995]
(a) sin x c (b) cos x c (c) e log cos x c (d) None of these
e
m log x
24. The value of dx is
x m 1 e m log x em em
(a) k (b) k (c) k (d) k
m 1 m log x x
e 5 log x e 4 log x
25.
e 3 log x
e 2 log x
dx [MNR 1985]
x3
(a) e . 3 3 x c (b) e 3 log x c (c) c (d) None of these
3
1 5
26. If f ' (x ) x and f (1) , then f (x )
x 2
x2 x2 x2 x2
(a) log x 2 (b) log x 1 (c) log x 2 (d) log x 1
2 2 2 2
27.
1 sin x dx [MP PET 1995]
1 x x 1 x x
(a) sin cos c (b) sin cos c (c) 2 1 sin x c (d) 2 1 sin x c
2 2 2 2 2 2
x
28. 1 sin dx [IIT 1980; MP PET 1989]
2
1 x x x x x x x x
(a) cos sin c (b) 4 cos sin c (c) 4 sin cos c (d) 4 sin cos c
4 4 4 4 4 4 4 4 4
29.
1 sin 2 x dx .......... ......, x (0, / 4 ) [MP PET 1987]
(a) sin x cos x (b) sin x cos x (c) tan x sec x (d) sin x cos x
dx
30. [MP PET 1991]
1 sin x
(a) x cos x c (b) 1 sin x c (c) sec x tan x c (d) sec x tan x c
cos x 1
31.
cos x 1
dx [MP PET 1989, 92]
x 1 x 1 x x
(a) 2 tan x c (b) tan x c (c) x tan c (d) x 2 tan c
2 2 2 2 2 2
32.
1 cos x dx equals [Rajasthan PET 1996]
x x x x
(a) 2 2 sin c (b) 2 2 sin c (c) 2 2 cos c (d) 2 2 cos c
2 2 2 2
dx
33. [MP PET 1990]
x x 2
(a)
1 3/2
3
x
(x 2) c (b)
3
2 3/2
x
(x 2) c (c)
1
3
(x 2) x c (d)
2
3
(x 2) x c
, Indefinite Integral 289
dx
34. [AISSE 1989]
x a x b
(a)
2
3(b a)
(x a) (x b) c (b)
2
3(a b)
(x a) (x b) c
(c)
2
3(a b)
(x a) (x b) c (d) None of these
3x3 2 x
35.
x
dx [Roorkee 1976]
(a) x 3 x c (b) x 3 x c (c) x 3 2 x c (d) x 3 4 x c
5(x 6 1)
36.
x2 1
dx
5 3
(a) 5(x 7 x ) tan 1 x c (b) x 5 x 5x c
3
(c) 3 x 4 5 x 2 15 x c (d) 5 tan 1 (x 2 1) log( x 2 1) c
tan x cot x
dx
37. [MP PET 1991]
cos 2 x sin 2 x sin 2 x cos 2 x
(a) c (b) c (c) c (d) c
4 4 4 4
1
38.
2 sin x x dx is equal to [MP PET 1999]
1 1
(a) 2 cos x log x C (b) 2 cos x log x C (c) 2 sin x C (d) 2 cos x C
x2 x2
39.
sin 2 x cos 3 x dx [Roorkee 1976]
1 1 1 1 1 1
(a) cos x cos 5 x c (b) cos x cos 5 x c (c) cos x cos 5 x c (d) cos x cos 5 x c
2 5 2 5 5 5
(sin 2 x cos 2 x)dx
1
40. If sin(2 x a) b, then [Roorkee 1978; MP PET 2001]
2
5 5
(a) a ,b 0 (b) a ,b 0 (c) a , b any constant (d) a , b any constant
4 4 4 4
(sin 2 x cos 2 x ) dx
1
41. If sin(2 x c) a, then the value of a and c is [Roorkee 1978]
2
(a) c and a k (an arbitrary constant) (b) c and a
4 4 2
(c) c and a is an arbitrary constant (d) None of these
2
sin 5 x cos 3 x dx
cos 8 x
42. If A, then A= [MP PET 1992]
16
sin 2 x cos 2 x
(a) constant (b) constant (c) Constant (d) None of these
16 4
43. If
2 1 sin x dx 4 cos(ax b) C then the value of (a,b) is [UPSEAT 2002]
1
(a) , (b) 1, (c) 1,1 (d) None of these
2 4 2
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