Q102 / 200 · TS EAMCET 2001 · Mathematics

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

  • Correct answer
    A
    $\frac{1}{a b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)+C$
  • B
    $\tan ^{-1}\left(\frac{a \tan x}{b}\right)+C$
  • C
    $\frac{1}{a b} \tan ^{-1}\left(\frac{b \tan x}{a}\right)+C$
  • D
    $\tan ^{-1}\left(\frac{b \tan x}{a}\right)+C$

Solution

$\begin{aligned} & \int \frac{d x}{a^2 \sin ^2 x+b^2 \cos ^2 x} \\ & =\frac{1}{a^2} \int \frac{\sec ^2 x d x}{\left(\frac{b}{a}\right)^2+\tan ^2 x} \\ & \Rightarrow \quad=\frac{1}{a b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)+C\end{aligned}$

Hence, the correct answer is option: 'A'.

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