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Evaluate int(0)^(1) (sin^(-1)x)/(sqrt(1-...

Evaluate `int_(0)^(1) (sin^(-1)x)/(sqrt(1-x^(2)))dx`.

Text Solution

Verified by Experts

`"Let sin"^(-1) x=t`
`rArr " "(1)/(sqrt(1-x^(2)))dx =dt`
`At x=0, " "t=sin^(-1) 0=0`
`At x=1 " "t=sin^(-1) 1=pi//2`
`:. int_(0)^(1) (sin^(-1)x)/(sqrt(1-x^(2))) dx =int_(0)^(pi//2) t dt = [(t^(2))/(2)]_(0)^(pi//2)`
`=1/2 ((pi)/(2))^(2) -0=(pi^(2))/(8).`
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