Q: _0^ d x (x^2+a^2 ) (x^2+b^2 ) is

Let aligned I & = _0^ 1 (x^2+a^2 ) (x^2+b^2 ) d x \\ I & = 1 a^2-b^2 _0^ (x^2+a^2 )- (x^2+b^2 ) (x^2+a^2 ) (x^2+b^2 ) d x \\ I & = 1 a^2-b^2 _0^ 1 (x^2+b^2 ) - 1 (x^2+a^2 ) d x \\ I & = 1 a^2-b^2 [ 1 b ^ -1 x b - 1 a ^ -1 x a ]_0^ \\ I & = 1 a^2-b^2 [ 1 b 2 - 1 a 2 ] \\ I & = 1 (a+b)(a-b) [ a-b a b ] 2 \\ I & = 2 a b(a+b) aligned

Question 1

_0^ d x (x^2+a^2 ) (x^2+b^2 ) is

Accepted Answer

Let aligned I & = _0^ 1 (x^2+a^2 ) (x^2+b^2 ) d x \ I & = 1 a^2-b^2 _0^ (x^2+a^2 )- (x^2+b^2 ) (x^2+a^2 ) (x^2+b^2 ) d x \ I & = 1 a^2-b^2 _0^ 1 (x^2+b^2 ) - 1 (x^2+a^2 ) d x \ I & = 1 a^2-b^2 [ 1 b ^ -1 x b - 1 a ^ -1 x a ]_0^ \ I & = 1 a^2-b^2 [ 1 b 2 - 1 a 2 ] \ I & = 1 (a+b)(a-b) [ a-b a b ] 2 \ I & = 2 a b(a+b) aligned

Question 2

The value of definite integral _0^ / 2 ( x) d x is

Accepted Answer

Let I= _0^ 2 ( x) d x....(i) By applying property, aligned _a^b f(x) d x & = _a^b f(a+b-x) d x \ I & = _0^ 2 ( ( 2 -x ) ) d x \ I & = _0^ 2 ( x) d x (ii) aligned Adding Eqs. (i) and (ii), aligned & 2 I= _0^ 2 [ ( x)+ ( x)] d x \ & 2 I= _0^ 2 1 d x=0 aligned

Question 3

_5^9 3 x^2 3 x^2+ (588-84 x+3 x^2 ) d x is equal to

Accepted Answer

Let I= _5^9 3 x^2 d x 3 x^2+ (588-84 x+3 x^2 ) ...(i) aligned & = _5^9 3 x^2 d x 3 x^2+ 3(14-x)^2 \ & = _5^9 3(14-x)^2 d x 3(14-x)^2+ 3(14-(14-x))^2 \ & [ _a^b f(x) d x= _a^b f(a+b-x) d x ] aligned I= _5^9 3(14-x)^2 d x 3(14-x)^2+ 3 x^2 ...(ii) Adding Eqs. (i) and (ii), we get aligned & 2 I= _5^9 3 x^2+ 3(14-x)^2 3(14-x)^2+ 3 x^2 d x \ & 2 I = _5^9 d x=9-5=4 I=2 aligned

Question 4

The integral x^2 (x ^2 x+ x ) (x x+1)^2 d x is equal to

Accepted Answer

(c) We note that d d x (x x+1)=x ^2 x+ x integrating by parts with x^2 as first function, we get aligned & I= x^2 x ^2 x+ x (x x+1)^2 d x \ & =x^2 (- 1 x x+1 )- 2 x (- 1 x x+1 ) d x \ & =- x^2 x x+1 +2 x x x x+ x d x \ & =- x^2 x x+1 +2 _e|x x+ x|+c \ & ( d d x (x x+ x)=x x ) aligned

Question 5

_ n _ r=3 ^ r^3-8 r^3+8 equals to

Accepted Answer

aligned & _ n _ r=3 ^ r^3-8 r^3+8 = ( 3^3-8 3^3+8 ) ( 4^3-8 4^3+8 ) \ & ......... ( n^3-8 n^3+8 ) aligned array r = _ n [ (3-2) (3^2+2^2+3.2 ) (3+2) (3^2+2^2-3.2 ) ] [ (4-2) (4^2+2+4.2 ) (4+2) (4^2+2^2-4.2 ) ] \ . . [ (n-2) (n^2+2^n+n .2 ) (n+2) (n^2+2^2-n .2 ) ] array gathered = _ n [ (3-2)(4-2) (n-2) (3+2)(4+2) (n+2) ] \ [ (3^2+2^2+3.2 ) (4^2+2^2+4.2 ) . (n^2+2^2+n .2 ) (3^2+2^2-3.2 ) (4^2+2^2-4.2 ) . . (n^2+2^2-n .2 ) ] gathered = [ 1 2 3.4 .5 .6 . 5 6.7 .8 . . ] [ 19 28 39.52 .63 . . 7 12

Question 6

_0^ 2 ( 4 +x )+ ( 3 4 +x ) x+ x d x=

Accepted Answer

Let I= _0^ 2 ( 4 +x )+ ( 3 4 +x ) x+ x d x I= _0^ / 2 ( 3 4 -x )+ ( 5 4 -x ) x+ x d x I= _0^ / 2 ( 4 +x )- ( 3 4 +x ) x+ x d x Now I+I= _0^ / 2 2 ( 4 +x ) x+ x d x I= _0^ / 2 1 2 x+ 1 2 x x- x d x= 2 2

Question 7

_0^ / 4 x e^ x d x is equal to

Accepted Answer

Suppose, I= _0^ / 2 x e^ x d x Let x=t x d x=d t x 0 t 0 and x 2 t 1 So I= _0^1 e^t d t= [e^t ]_0^1=e-1

Question 8

_ -a ^ 4 (x^ 8 -x^ 4 +x^ 2 +1 ) d x=2 _ 0 ^ 4 (x^ 8 -x^ 4 +x^ 2 +1 ) d x then a=

Accepted Answer

If f (x) is an even function then _ -a ^ a f(x) d x=2 _ 0 ^ a f(x) d x Here f (x)=x^ 8 -x^ 4 +x^ 2 +1 is an even function, therefore a=4.

Question 9

Evaluate: _ 0 ^ 1 5+4 x d x

Accepted Answer

We have, _ 0 ^ 1 5+4 x d x = _ 0 ^ 1 5+4 ( 1- ^ 2 x 2 1+ ^ 2 x 2 ) d x = _ 0 ^ 1+ ^ 2 x 2 5 (1+ ^ 2 x 2 )+4 (1- ^ 2 x 2 ) d x = _ 0 ^ 1+ ^ 2 x 2 9+ ^ 2 x 2 dx = _ 0 ^ ^ 2 x 2 9+ ^ 2 x 2 dx Let x 2 =t 1 2 ^ 2 x 2 d x=d t Also, x=0 t=0 and x= t= I = _ 0 ^ dt 9+t^ 2 I =2 _ 0 ^ dt 9+t^ 2 I = 2 3 [ ^ -1 t 3 ]_ 0 ^ = 2 3 [ ^ -1 - ^ -1 0 ] = 2 3 ( 2 -0 )= 3

Question 10

_ 0 ^ dx ( x ^ 2 + a ^ 2 ) ( x ^ 2 + b ^ 2 ) is

Accepted Answer

_ 0 ^ d x (x^ 2 +a^ 2 ) (x^ 2 +b^ 2 ) = 1 b^ 2 -a^ 2 _ 0 ^ (x^ 2 +b^ 2 )- (x^ 2 +a^ 2 ) (x^ 2 +a^ 2 ) (x^ 2 +b^ 2 ) = 1 b^ 2 -a^ 2 _ 0 ^ [ 1 x^ 2 +a^ 2 - 1 x^ 2 +b^ 2 ] d x = 1 b^ 2 -a^ 2 [ 1 a ^ -1 x a - 1 b ^ -1 x b ]_ 0 ^ = 1 b^ 2 -a^ 2 [ 2 a - 2 b ]= 2 a b(a+b)

Question 11

_ / 3 ^ / 2 x ( [x]-x) dx is equal to

Accepted Answer

In the interval 3 to 2 ,[ x ]=1 aligned I &= _ / 3 ^ / 2 x ( - x ) dx = _ / 3 ^ / 2 x x d x \ &=[- x x + x ]_ / 3 ^ / 2 =1- 3 2 + 6 aligned

Definite Integration — Previous Year Questions & Solutions