$\int \frac{\sin 2 x}{a^{2} + b^{2} \sin^{2} x}$

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Solution

$Let I = \int \frac{\sin 2 x}{a^{2} + b^{2} \sin^{2} x} d x Putting a + b^{2} \sin^{2} x = t \Rightarrow b^{2} (2 \sin x \cos x) d x = d t \Rightarrow b^{2} \times \sin 2 x d x = d t ∴ I = \frac{1}{b^{2}} \int \frac{d t}{t} = \frac{1}{b^{2}} \ln |t| + C = \frac{1}{b^{2}} \ln |a^{2} + b^{2} \sin^{2} x| + C (∵ t = a + b^{2} \sin^{2} x)$

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