First slide

Methods of integration

Question

$\int \frac{d x}{a^{2} \cos^{2} x + b^{2} \sin^{2} x} (a, b > 0)$ is equal to

Easy

A

$\sin^{- 1} ((a / b) \tan x) + C$
B

$\tan^{- 1} ((b / a) \tan x) + C$
C

$\frac{1}{a b} \tan^{- 1} ((\frac{b}{a}) \tan x) + C$
D

none of these

Solution

Putting $t = \tan x$ we get
$\begin{array}{l} \int \frac{d x}{a^{2} \cos^{2} x + b^{2} \sin^{2} x} = \int \frac{\sec^{2} x d x}{a^{2} + b^{2} \tan^{2} x} = \int \frac{d t}{a^{2} + b^{2} t^{2}} \\ = \frac{1}{b^{2}} \int \frac{d t}{t^{2} + (a^{2} / b^{2})} = \frac{1}{b^{2}} \cdot \frac{b}{a} \tan^{- 1} (\frac{b}{a} t) + C \\ = \frac{1}{a b} \tan^{- 1} (\frac{b}{a} \tan x) + C . \end{array}$

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