Integration of f(x)=1+x2 with respect to x2, is - Infinity Learn by Sri Chaitanya

A
$\frac{2}{3} \frac{{(1 + x^{2})}^{3 / 2}}{x} + C$
B
$\frac{2}{3} {(1 + x^{2})}^{3 / 2} + C$
C
$\frac{2 x}{3} {(1 + x^{2})}^{3 / 2} + C$
D
none of these

Solution:

Let

$\begin{array}{l} I = \int f (x) d (x^{2}) = \int \sqrt{1 + x^{2}} d (x^{2}) \\ \Rightarrow I = \int \sqrt{1 + x^{2}} d (1 + x^{2}) [∵ d (x^{2}) = d (1 + x^{2})] \\ \Rightarrow I = \frac{2}{3} {(1 + x^{2})}^{3 / 2} + C \end{array}$

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