Primitive of $f (x) = x {.2}^{I n (x^{2} + 1)}$ with respect to x is

\frac{2^{I n (x^{2} + 1)}}{(x^{2} + 1)} + C

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\frac{(x^{2} + 1) 2^{I n (x^{2} + 1)}}{I n 2 + 1} + C

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\frac{(x^{2} + 1)^{I n 2 + 1}}{2 (I n 2 + 1)} + C

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\frac{(x^{2} + 1)^{I n 2}}{2 (I n 2 + 1)} + C

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Solution

The correct option is C $\frac{(x^{2} + 1)^{I n 2 + 1}}{2 (I n 2 + 1)} + C$
$I = \int x 2^{I n (x^{2} + 1)} d x$ 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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