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Evaluate: int(e^tan^((-1)x))/((1+x^2))[(...

Evaluate: `int(e^tan^((-1)x))/((1+x^2))[(sec^(-1)sqrt(1+x^2)+cos^(-1)((1-x^2)/(1+x^2))]dx(x >0)dot`

Text Solution

Verified by Experts

For `x gt 0,sec^(-1)sqrt(1+x^(2))=tan^(-1)x`
and `cos^(-1)((1-x^(2))/(1+x^(2)))=2tan^(-1)x`
` :. I=int(e^(tan^(-1)x))/((1+x^(2)))[(tan^(-1)x)^(2)+2tan^(-1)x]dx " "["Put "tan^(-1)x=t]`
`=int e^(t)(t^(2)+2t)dt`
`=e^(t)t^(2)+C=e^(tan^(-1)x)(tan^(-1)x)^(2)+C`
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