Primitive of f ( x ) = x .2 I n ( x 2 + 1 ) with respect to x is Options: A . &nbsp2In(x2+1)(x2+1)+C B . &nbsp(x2+1)2In(x2+1)In2+1+C C . &nbsp(x2+1)In2+12(In2+1)+C D . &nbsp(x2+1)In22(In2+1)+C

Answer: Option C : C I = ∫ x 2 I n ( x 2 + 1 ) d x let x 2 + 1 = t ; x d x = d t 2 Hence I = 1 2 ∫ 2 I n t d t = 1 2 ∫ t I n 2 d t = 1 2 . t I n 2 + 1 I n 2 + 1 + C = 1 2 . ( x 2 + 1 ) I n 2 + 1 I n 2 + 1 + C

Primitive of f(x)=x.2In(x2+1) with respect to x is

Question

Primitive of

f (x) = x {.2}^{I n (x^{2} + 1)}

with respect to x is

Options:

A . 2In(x2+1)(x2+1)+C

B . (x2+1)2In(x2+1)In2+1+C

C . (x2+1)In2+12(In2+1)+C

D . (x2+1)In22(In2+1)+C

Answer: Option C
:
C

I = \int x 2^{I n (x^{2} + 1)} d x

let

x^{2} + 1 = t; x d x = \frac{d t}{2}

Hence

I = \frac{1}{2} \int 2^{I n t} d t = \frac{1}{2} \int t^{I n 2} d t = \frac{1}{2} . \frac{t^{I n 2 + 1}}{I n 2 + 1} + C = \frac{1}{2} . \frac{(x^{2} + 1)^{I n 2 + 1}}{I n 2 + 1} + C

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