1.

The value of underset(0)overset(pi//2)int (sin^(3)x)/(sin x + cos x)dxis (pi-1)/(k) . The value of k is ________.

Answer»


Solution :`int_(0)^(2) (sin^(3)x)/(sin x+cosx)dx=I` (say) `RARR I= int_(0)^(pi//2) (COS^(3)x)/(sin x + cos x) dx`
`rArr 2I = int_(0)^(pi//2) ( sin^(3) x+cos^(3)x)/(sin x+ cos x) dx rArr 2I = int_(0)^(pi//2) ( sin^(2)x+cos^(2)x-sin x cos x) dx`
`rArr 2 I = int_(0)^(pi//2)(1-1/2 sin 2 x) dx rArr 2 I = x + (cos 2x)/(4) ]_(0)^(pi//2)`
`rArr 2I=(pi/2 - 1/4) - (1/4)= pi/4-1/2 rArr I=(pi-1)/(4)`


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