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The value of int(0)^(n//2)(sin^(3)x)/(si...

The value of `int_(0)^(n//2)(sin^(3)x)/(sinx+cos x)` dx is

A

`(pi-2)/4`

B

`(pi-2)/8`

C

`(pi-1)/4`

D

`(pi-1)/2`

Text Solution

Verified by Experts

The correct Answer is:
C
`I = int_(0)^(pi//2)(sin^(3)x)/(sinx+cosx)dx` ……..(1)
By `p 4 int_(0)^(a) fxdx = int_(0)^(a) f(a-x)dx`
`I = int_(0)^(pi//2) (cos^(3)x)/(cos x+sinx) dx` …….(2)
`2I = int_(0)^(pi//2)((sin^(3)x+cos^3x))/((sin x + cos x)) dx`
`= int_(0)^(pi//2) (sin^(2)x +cos^(2)x-sinxcosx)dx`
`int_(0)^(pi//2)(1-(sin2x)/(2))dx=(x+(cos 2x)/(4))_(0)^(pi//2)`
`2I = ((pi)/(2) + (-1)/(4))-(0+1/4) = (pi)/2 - 1/2`
`:. I = ((pi-1)/(4))`.
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