# Integrate : $\int \sqrt {\tan \theta} . d \theta$

$(a)\;\frac{1}{2} \tan ^{-1} \bigg(\frac{\sqrt {\tan \theta } -1/ \sqrt {\tan \theta }}{\sqrt 2}\bigg) -\frac{1}{2 \sqrt 2 } \log \bigg(\frac{\sqrt {\tan \theta}+\frac {1}{\sqrt {\tan \theta }}-\sqrt 2 }{\sqrt {\tan \theta}+ \frac{1}{\sqrt {\tan \theta}}+\sqrt 2 }\bigg)+c \\(b)\;\frac{5}{2} \tan ^{-1} \bigg(\frac{\sqrt {\tan \theta } -1/ \sqrt {\tan \theta }}{\sqrt 2}\bigg) -\frac{1}{2 \sqrt 2 } \log \bigg(\frac{\sqrt {\tan \theta}+\frac {1}{\sqrt {\tan \theta }}-\sqrt 2 }{\sqrt {\tan \theta}+ \frac{1}{\sqrt {\tan \theta}}+\sqrt 2 }\bigg)+c \\(c)\;\frac{1}{2} \tan ^{-1} \bigg(\frac{\sqrt {\tan \theta } +1/ \sqrt {\tan \theta }}{\sqrt 2}\bigg) -\frac{1}{2 \sqrt 2 } \log \bigg(\frac{\sqrt {\tan \theta}+\frac {1}{\sqrt {\tan \theta }}-\sqrt 2 }{\sqrt {\tan \theta}+ \frac{1}{\sqrt {\tan \theta}}+\sqrt 2 }\bigg)+c \\ (d)\;None$

$\tan \theta= t^2$
$\sec^2 \theta. d \theta = 2t.dt$
=> $d \theta = \large\frac{2tdt}{1+t^2}$
=> $\int \large\frac{2(t)^2}{1+t^4}$$dt => \int \large\frac{2t^2}{1+t^4}$$dt$
=> $\int \large\frac{t^2+1}{1+t^4}$$dt+\int \large\frac{t^2-1}{1+t^4}$$dt$
$\large\frac{1}{2}$$\tan ^{-1} \bigg(\frac{1-1/t } {\sqrt 2} \bigg) -\frac{1}{2 \sqrt 2 }$$ \log \bigg(\frac{t+1/t -\sqrt 2}{t+ 1/t+\sqrt 2}\bigg)+c$