The value of indefinite integral: $ \int \sqrt{\frac{\sin x-\sin ^{3} x}{1-\sin ^{3} x}} d x $ equals: (where $ c $ is constant of integration)

Question

The value of indefinite integral: $ \int \sqrt{\frac{\sin x-\sin ^{3} x}{1-\sin ^{3} x}} d x $ equals: (where $ c $ is constant of integration), $ \frac{3}{2} \cos ^{-1}\left(\sin ^{\frac{3}{2}} x\right)+c $, $ \frac{2}{3} \sin ^{-1}\left(\sin ^{\frac{3}{2}} x\right)+c $, $ \frac{3}{2} \sin ^{-1}\left(\sin ^{\frac{3}{2}} x\right)+c $, $ \frac{2}{3} \tan ^{-1}\left(\sin ^{\frac{3}{2}} x\right)+c $

MARKS App · Accepted Answer

Let I = ∫ sin x - sin 3 x 1 - sin 3 x d x ⇒ I = ∫ sin x 1 - sin 2 x 1 - sin 3 x d x ⇒ I = ∫ sin x cos x 1 - sin 3 x d x Put s i n 3 2 x = t ⇒ 3 2 sin x cos x d x = d t ⇒ I = 2 3 ∫ d t 1 - t 2 ⇒ I = 2 3 s i n - 1 t + c ⇒ I = 2 3 sin - 1 sin 3 2 x + c

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