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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)))^(2)+"cos"^(-1)((1-x^(2))/(1+x^(2)))]dx(x gt 0)`.

A

`I=e^(tan^(-1)x)(tan^(-1)x)+C`

B

`I=e^(tan^(-1)x)(sec^(-1)sqrt(1+x^(2)))^(2)+C`

C

`I=(1)/(2)e^(tan^(-1)x)(tan^(-1)x)^(2)+C`

D

`I=e^(tan^(-1)x)("cosec"^(-1)sqrt(1+x^(2)))^(2)+C`

Text Solution

Verified by Experts

The correct Answer is:
b
Let `tan^(-1)x=thetaorx=tantheta` . Then,
`I=inte^(theta)(theta^(2)+2 theta)d theta=e^(theta)theta^(2)+C`
`rArrI=e^(tan^(-1)x)(tan^(-1)x)^(2)+C`
`rArrI=e^(tan^(-1)x)(sec^(-1)sqrt(1+x^(2)))^(2)+C`
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