Q: The value of ( x) d x is :

Let $ I = ( x ) 1 ~d x . . (i) $ Use integral by parts, $ aligned &I= ( x) x- [- ( x)] 1 x x d x \\ &=x ( x)+ ( x) 1 d x \\ &=x ( x)+ [ ( x) x- ( x) 1 x x d x ]+C \\ &=x ( x)+ [x ( x)- ( x) d x ]+C \\ &=x\ ( x)+ ( x)\ -I+C[f r o m Eq. (i) ] \\ & I= x 2 \ ( x)+ ( x)\ +C aligned $

Q: If 3 x+1 (x-3)(x-5) d x= -5 (x-3) d x+ B (x-5) d x then the value of B is

We have, array l 3 x+1 (x-3)(x-5) = -5 x-3 + B x-5 \\ 3 x+1=-5(x-5)+ B (x-3) \\ Put x=5 \\ 3(5)+1= B (5-3) \\ 16=2 ~B or B =8 array

Question 1

The value of e^ ( - ) d is

Accepted Answer

Let I= e^ ( - ) d Put =t ^2 d =d t d = d t 1+t^2 I= e^t ( 1+t^2 - t 1+t^2 ) d t 1+t^2 = e^t ( 1 1+t^2 - t (1+t^2 )^ 3 / 2 ) d t Integrating first part by parts we have, = 1 1+t^2 e^t+ t (1+t^2 )^ 3 / 2 e^t d t - t (1+t^2 )^ 3 / 2 e^t d t+c = e^t 1+t^2 +c=e^ +c

Question 2

If e ^ x (1+ x ) dx 1+ x = e ^ x f ( x )+ C , then f ( x ) is equal to

Accepted Answer

e^x (1+ x) (1+ x) d x = e^x [ 1 2 ^2 x 2 + x 2 ] d x = 1 2 e^x ^2 x 2 d x+ e^x x 2 d x = e ^ x x 2 + C But I = e ^ x f ( x )+ C (given) f ( x )= x 2

Question 3

x+3 (x+4)^2 e^x d x is equal to

Accepted Answer

Suppose, I= x+3 (x+4)^2 e^x d x= (x+4) 1 (x+4)^2 e^x d x = e^x (x+4) - e^x (x+4)^2 d x = e^x ( 1 (x+4) - 1 (x+4)^2 ) d x =e^x 1 (x+4) +C [ e^x \ f(x)+f^ (x) \ d x=e^x f(x)+C ]

Question 4

The value of 1 [(x-1)^3(x+2)^5 ]^ 1 4 d x, is

Accepted Answer

I= 1 ( x-1 x+2 )^ 3 / 4 (x+2)^2 d x Let x-1 x+2 =t 3 d x (x+2)^2 =d t aligned I & = 1 3 1 t^ 3 / 4 d t= 1 3 ( t^ 1 / 4 1 / 4 )+C \ & = 4 3 t^ 1 / 4 +C= 4 3 ( x-1 x+2 )^ 1 / 4 +C aligned

Question 5

Let f(x)= x^2 d x (1+x^2 ) (1+ .1+x^2 ) . and f(0)=0, then the value of f(1) be

Accepted Answer

f(x)= x^2 d x (1+x^2 ) (1+ 1+x^2 ) Let x= d x= ^2 d = (1+x^2 ) d f(x)= x^2 d x (1+x^2 ) (1+ 1+x^2 ) = ^2 ^2 d ^2 (1+ ) = ^2 d 1+ = ^2 d (1+ ) = 1- ^2 d (1+ ) = (1- ) d = d - d = (x+ 1+x^2 )- ^ -1 x+C f(0)= (0+ 1+0 )- ^ -1 (0)+C 0= 1-0+C C=0 f(1)= (1+ 1+1^2 )- ^ -1 (1) = (1+ 2 )- 4

Question 6

If x (x- ) d x = A x+ B (x- )+ C , then value of (A, B) is

Accepted Answer

x (x- ) dx = (x- + (x- ) dx = (x- ) + (x- ) (x- ) d x = \ + (x- )\ d x =( ) x+( ) (x- )+ C A = , B =

Question 7

The value of ( x) d x is :

Accepted Answer

Let $ I = ( x ) 1 ~d x . . (i) $ Use integral by parts, $ aligned &I= ( x) x- [- ( x)] 1 x x d x \ &=x ( x)+ ( x) 1 d x \ &=x ( x)+ [ ( x) x- ( x) 1 x x d x ]+C \ &=x ( x)+ [x ( x)- ( x) d x ]+C \ &=x\ ( x)+ ( x)\ -I+C[f r o m Eq. (i) ] \ & I= x 2 \ ( x)+ ( x)\ +C aligned $

Question 8

e^ x (1+x) ^ 2 (e^ x x ) dx equals

Accepted Answer

e^ x (1+x) ^ 2 (e^ x x ) dx Let x e^ x =t (x e^ x +e^ x )= dt dx d x= dt e^ x (x+1) e^ x (1+x) ^ 2 (e^ x x ) dx = e^ x (1+x) ^ 2 t d t e^ x (1+x) = 1 ^ 2 t dt = ^ 2 t dt = t+ C = (x e^ x )+ C

Question 9

If 3 x+1 (x-3)(x-5) d x= -5 (x-3) d x+ B (x-5) d x then the value of B is

Accepted Answer

We have, array l 3 x+1 (x-3)(x-5) = -5 x-3 + B x-5 \ 3 x+1=-5(x-5)+ B (x-3) \ Put x=5 \ 3(5)+1= B (5-3) \ 16=2 ~B or B =8 array

Indefinite Integration — Previous Year Questions & Solutions