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int(0)^(pi/2) (sinx)/(1+cos^(2)x)dx...

`int_(0)^(pi/2) (sinx)/(1+cos^(2)x)dx`

Text Solution

Verified by Experts

`"Let " I= int_(0)^(pi//2)(sin x)/(1+cos^(2)x)dx`
` "Let " cos x=t`
`rArr sin x dx =dt rArr t=cos 0=1`
`" and "x=(pi)/(2) rArr t=cos .(pi)/(2)=0`
`:. I= int_(1)^(0)(1)/(1+t^(2))(-dt) =- int_(1)^(0)(1)/(1+t^(2))dx`
`=-[tan^(-1)t]_(1)^(0)`
`=-[tan^(-1)(0)-tan^(-1)(1)]`
`=-(0-(pi)/(4))=(pi)/(4)`
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