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Evaluate the integrals int0 1sin^(-1)((2...

Evaluate the integrals `int0 1sin^(-1)((2x)/(1+x^2))dx`

Text Solution

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`int sin^(-1) .(2x)/(1+x^(2)) dx=2 int tan^(-1) x. 1dx`
` " Let " 1+x^(2) =t rArr 2x dx =dt rArr xdx= (dt)/(2)`
` =2 [tan^(-1) x.int 1dx -int {(d)/(dx) tan^(-1) x int 1dx } dx]`
`=2[tan^(-1) x- int (x)/(1+x^(2))dx]`
`=2 [x tan^(-1) x-int (dt)/(2t)]`
`=2x tan^(-1) x-log t+C`
`=2x tan^(-1) x-log (1+x^(2))+C`
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