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Q.

$\int \frac{dx}{a^{2} \sin^{2} x + b^{2} \cos^{2} x}$ is equal to

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a

$\frac{1}{ab} \tan^{- 1} (\frac{atan x}{b}) + C$

b

None of the above

c

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

d

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

answer is A.

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Detailed Solution

$\int \frac{dx}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} = \int \frac{\sec^{2} xdx}{a^{2} \tan^{2} x + b^{2}} = \frac{1}{a^{2}} \int \frac{dt}{t^{2} + {(\frac{b}{a})}^{2}}$

$where, \begin{aligned} t = \tan x = \frac{1}{a^{2}} \cdot \frac{a}{b} \tan^{- 1} \frac{t}{b / a} + C \\ = \frac{1}{ab} \tan^{- 1} \frac{a (\tan x)}{b} + C \end{aligned}$

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