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What is $\int \frac{\mathrm{dx}}{\mathrm{a}^{2} \sin ^{2} \mathrm{x}+\mathrm{b}^{2} \cos ^{2} \mathrm{x}}$ equal to?
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Verified Answer
The correct answer is:
$\mathrm{c}+\frac{1}{\mathrm{ab}} \tan ^{-1}\left(\frac{\operatorname{atan} \mathrm{x}}{\mathrm{b}}\right)$
$\mathrm{I}=\frac{1}{a^{2}} \int \frac{\sec ^{2} x}{\tan ^{2} x+\left(\frac{b}{a}\right)^{2}}$
$=\frac{1}{a^{2}} \times \frac{a}{b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)+c$
$=c+\frac{1}{a b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)$
$=\frac{1}{a^{2}} \times \frac{a}{b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)+c$
$=c+\frac{1}{a b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)$
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