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Question: Value of I =\(\int_{0}^{\pi/2}\frac{\sin^{3}x\cos x}{\sin^{4}x + \cos^{4}x}\) dx is equal to...

Value of I =∫0π/2sin⁡3xcos⁡xsin⁡4x+cos⁡4x\int_{0}^{\pi/2}\frac{\sin^{3}x\cos x}{\sin^{4}x + \cos^{4}x} dx is equal to

A

p/8

B

p/4

C

p/2

D

p

Answer

p/4

Explanation

Solution

I =∫0π/2cos⁡3xsin⁡xsin⁡4x+cos⁡4x\int_{0}^{\pi/2}\frac{\cos^{3}x\sin x}{\sin^{4}x + \cos^{4}x}dx

Ž I + I =∫0π/2sin⁡xcos⁡xsin⁡4x+cos⁡4x\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x + \cos^{4}x}dx

put sin2x = t