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int(0)^(pi//2)(sin^(3//2)x)/(sin^(3//2)x...

`int_(0)^(pi//2)(sin^(3//2)x)/(sin^(3//2)x+cos^(3//2)x)dx`

Text Solution

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`"Let "I= int_(0)^(pi//2) (sin^(3/2)x)/(sin^(3/2)x+cos^(3/2)x)dx" "......(1)`
`rArr I=int_(0)^(pi//2) (sin ^(3/2)((pi)/(2)-x))/(sin^(3/2)((pi)/(2)-x)+cos^(3/2)((pi)/(2)-x))dx`
`[ :' int_(0)^(a) f(x) dx= int_(0)^(a) f(a-x)dx]`
`=int_(0)^(pi//2)(cos^(3/2)x)/(cos^(3/2)x+sin^(3/2)x)dx " "......(2)`
Adding equations (1) and (2)
`2I= int_(0)^(pi//2) (sin^(3/2)x+cos^(3/2)x)/(sin^(3/2)x+cos^(3/2)x)dx`
`=int_(0)^(pi//2) 1dx =[x]_(0)^(pi//2) =(pi)/(2)-0`
`rArr " "I=(pi)/(4)`
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