Question

Integrate sinx cosx/a^2sin^2x + b^2cos^2x

Answer 1

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b^2cos^2x

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Please find below the solution to the asked query :

Let I = ∫sin x . cos xa2sin2x + b2cos2x dx=∫sin x . cos xa2sin2x + b2−b2sin2x dx=∫sin x . cos x(a2−b2) sin2x + b2 dxput sin2x = t⇒2 sin x . cos x dx = dt⇒sin x . cos x dx = dt2Now, I = 12∫dt(a2−b2)t2 + b2=12(a2−b2) ∫dtt2 + b2a2−b2=12(a2−b2)∫dtt2 + ⎛⎝ba2−b2√⎞⎠2=12(a2−b2)×a2−b2√b tan−1[ta2−b2√b] + C=12b×1a2−b2√ tan−1[a2−b2√ sin2xb] + C