Need questions related to integration ? Here it is strong your topic ASAP

®




MATHEMATICS
JEE (MAIN+ADVANCED)
ENTHUSIAST COURSE




EXERCISE
Indefinite & Definite Integration
English Medium

,JEE-Mathematics

EXERCISE # O-I
Straight Objective Type

1. Let g (x) be an antiderivative for f (x). Then ln 1 + ( g(x) ) ( 2
) is an antiderivative for
2f (x)g(x) 2f (x)g(x) 2f (x)
(A) 2
(B) 2
(C) 2
(D) none
1 + ( f (x) ) 1 + ( g(x) ) 1 + ( f (x) )
II0001
1 1
2. A function y = f (x) satisfies f "(x) = – 2
– π2 sin(πx) ; f '(2) = π+ and f (1) = 0. The value of
x 2
1
f   is
2
π
(A) ln 2 (B) 1 (C) – ln 2 (D) 1 – ln 2
2




N
II0002
x2
3. Consider ƒ(x) = ; g(t) = ∫ ƒ(t)dt. If g(1) = 0 then g(x) equals -
1 + x3
1 1  1 + x3  1  1 + x3  1  1 + x3 
(A) n(1 + x 3 )
E (B) n   (C) n   (D) n  
3 3  2  2  3  3  3 
II0003
2
x (1 − ln x)
4. ∫ ln 4 x − x 4
dx equals
LL
1 x 1 1 ln x − x 1  ln x 
(A) ln − ln ln 2 x − x 2 + C (B) ln − tan −1  +C
2 ln x 4 4 ln x + x 2  x 
1 ln x + x 1  ln x  1 ln x − x  ln x  
(C) ln + tan −1  +C (D)  ln + tan −1   + C
4 ln x − x 2  x  4 ln x + x  x 
II0004
(2x + 3) dx 1
5. If ∫ x(x + 1)(x + 2)(x + 3) + 1 = C –
f (x)
, where ƒ(x) is of the form of ax2 + bx + c then
A

(a + b + c) equals
(A) 4 (B) 5 (C) 6 (D) none
II0005
x3
6. ∫ dx is equal to -
( 2x + 1)
2 3


−2 −2
1 1  1 1 
(A) 2+ 2  +C (B) −  2 + 2  + C
4 x  4 x 
−2 2
1 1  1 1 
(C)  2 + 2  + C (D)  2 + 2  + C
2 x  4 x 
(where 'C' is integration constant)
II0006

54 E

, Indefinite Integration

3x 4 − 1
7. Primitive of w.r.t. x is -
(x 4 + x + 1) 2
x x x +1 x +1
(A) 4 +c (B) − 4 +c (C) 4
+c (D) − 4
+c
x + x +1 x + x +1 x + x +1 x + x +1
II0007
2
x +x
8. ∫ (ex + x + 1)2 dx equals
1 1
(A) C − l n (1 + (x + 1)e x ) − (B) C − l n 1 + (x + 1)e x +
1 + (x + 1)e x 1 + (x − 1)e x
1 1
(C) C − l n 1 − (x 2 − 1)e − x + (D) C − l n 1 + (x + 1)e − x −
1 + (x + 1)e x 1 + (x + 1)e − x
II0056
dx
9. ∫ equals




N
3
x (x + 1)7/2
5/2

1/6 −1/6 5/6 5/6
 x +1   x +1   x   x 
(A) −   +C (B) 6   +C (C)   +C (D) −   +C
 x   x   x +1   x +1 
II0009

10. ∫
1− x
E
x(1 + x 7 )
7
dx equals -

2
(A) ln |x| + ln |1 + x7| + c
7
2
(B) ln |x| −
2
7
1
ln |1 − x7| + c
LL
(C) ln |x| − ln |1 + x7| + c (D) ln |x| − ln |1 + x7| + c
7 7
II0010
11. Integral of 1 + 2 cot x(cot x + cosecx) w.r.t. x is
x x
(A) 2 ln cos +c (B) 2 ln sin +c
2 2
1 x
(C) ln cos + c (D) ln |sinx| – ln|cosecx – cotx| + c
2 2
A

II0011
x
e
12. ∫ x
( x + x ) dx equals
(A) 2e x
 x − x + 1 + C (B) e x
 x − 2 x + 1
   
(C) e x
(x + x ) + C (D) e x
(x + )
x +1 + C
II0012
2
2sin x − 1 cos x(2sin x + 1)
13. =
Let ƒ(x) + then ∫ e x ((ƒ(x) + ƒ '(x))dx
cos x 1 + sin x
(A) ex tanx + c (B) ex cotx + c (C) ex cosec2x + c (D) exsec2x + c
(where c is the constant of integration)
II0013

E 55

, JEE-Mathematics

x  x −3 
2
14. ∫  (x − 1)2  dx is equal to -
e

2
(x + 3)  x −3  x +1   1 
(A) e x +C (B) e x  +C (C) e x  +C (D) e x   +C
(x − 1)  x −1   x −1   x −1 
(where C is constant of integration)
II0014

(
ln x + 1 + x 2 ) dx equals -
15. ∫x . 1+ x 2

(
(A) 1 + x 2 ln x + 1 + x 2 − x + c ) (B)
x
2
(
. ln2 x + 1 + x 2 − ) x
1 + x2
+c


(C)
x
(
. ln2 x + 1 + x 2 + ) x
+c ( )
(D) 1 + x 2 ln x + 1 + x 2 + x + c




N
2 1+ x 2


II0015
p x p + 2q −1 − q x q −1
16. The evaluation of ∫ x 2p + 2q + 2x p+q +1 dx is



17.
(A) −
E xp
x p+q + 1

The integral ∫
+C


cot x e
(B)


sin x
xq
x p+q + 1
+C


cos x dx equals
(C) −
xq
x p+q + 1
+C (D)
xp
x p+q + 1
+C

II0017
LL
tan x e sin x sin x
(A) +C (B) 2e +C
cos x
1 sin x cot x e sin x
(C) − e +C (D) +C
2 2 cos x
II0018
−1 x
cot (e )
18. ∫ ex
dx is equal to -
A

1 cot −1 (e x ) 1 cot −1 (e x )
(A) n e 2x + 1 − +x+c (B) n e 2x + 1 + +x+c
2 ex 2 ex
1 cot −1 (e x ) 1 cot −1 (e x )
(C) n e 2x + 1 − −x+c (D) n e 2x + 1 + −x+c
2 ex 2 ex
II0019
∫ ( sin(101x)·sin )
99
19. x dx equals

sin(100x)(sin x)100 cos(100x)(sin x)100
(A) +C (B) +C
100 100
cos(100x)(cos x)100 sin(100x)(sin x)101
(C) +C (D) +C II0020
100 101


56 E