First slide

Methods of integration

Question

$\int \frac{1}{x^{2} \sqrt{1 + x^{2}}} d x is equal to$

Moderate

A

$\frac{- \sqrt{1 + x^{2}}}{x} + C$
B

$\frac{\sqrt{1 + x^{2}}}{x} + C$
C

$- x \sqrt{1 + x^{2}} + C$
D

$x \sqrt{1 + x^{2}} + C$

Solution

$Let I = \int \frac{1}{x^{2} \sqrt{1 + x^{2}}} d x = \int \frac{1}{x^{2} x \sqrt{1 + \frac{1}{x^{2}}}} d x$
$Put 1 + \frac{1}{x^{2}} = t \Rightarrow \frac{d x}{x^{3}} = \frac{- 1}{2} d t$
$\begin{array}{l} I = \int \frac{- 1}{2 \sqrt{t}} d t = - \sqrt{t} + C \\ = - \sqrt{1 + \frac{1}{x^{2}}} + C \\ = \frac{- \sqrt{x^{2} + 1}}{x} + C \end{array}$

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