First slide

Methods of integration

Question

Integration of $f (x) = 1 + x^{2}$ with respect to $x^{2}$ , is

Moderate

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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