First slide

Introduction to integration

Question

$Evaluate \int \frac{1}{x^{2} \sqrt{1 + x^{2}}} d x$

Easy

A

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

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

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

None of these

Solution

$L e t x = \tan θ o r d x = s e c^{2} θ d θ$
$\begin{array}{l} ∴ \int \frac{1}{x^{2} \sqrt{1 + x^{2}}} d x = \int \frac{\sec^{2} θ d θ}{\tan^{2} θ \sec θ} \\ = \int cosec θ \cot θ d θ \\ = - cosec θ + c \\ = \frac{- \sqrt{x^{2} + 1}}{x} + c \end{array}$

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