Evaluate the following integrals: ^ -1 ( x )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\tan^{-1}(\sqrt{\text{x}})\text{dx}$

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Answer

Let $\text{I}=\int\tan^{-1}(\sqrt{\text{x}})\text{dx}$Let $\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\text{I}=\int2\text{t}\tan^{-1}\text{t dt}$
$=2\Big[\tan^{-1}\text{t}\int\text{t dt}-\int\Big(\frac{1}{1+\text{t}^2}\int\text{t dt}\Big)\text{dt}\Big]$
$=2\Big[\frac{\text{t}^2}{2}\tan^{-1}\text{t}-\int\frac{\text{t}^2}{2(1+\text{t}^2)}\text{dt}\Big]$
$=\text{t}^2\tan^{-1}\text{t}-\int\frac{\text{t}^2+1-1}{1+\text{t}^2}\text{dt}$
$=\text{t}^2\tan^{-1}\text{t}-\int\Big(1-\frac{1}{1+\text{t}^2}\Big)\text{dt}$
$=\text{t}^2\tan^{-1}\text{t}-\text{t}+\tan^{-1}\text{t + C}$
$=(\text{t}^2+1)\tan^{-1}\text{t}-\text{t + C}$
$\text{I}=(\text{x}+1)\tan^{-1}\sqrt{\text{x}}-\sqrt{\text{x}}+\text{C}$

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