Question

Find the integral

Answer 1

\[\large\rm \int\limits \sqrt{\tan \theta}(\sec^2\theta)(\sec^2\theta)d \theta\quad=\quad \int\limits \sqrt{\tan \theta}(1+\tan^2\theta) \sec^2\theta~d \theta\]Ok with that first step?

Answer 2

Building on zepdrix's work:

Let \[\tan \theta = u => du = \sec^2 \theta d \theta\]

\[\int\limits_{}^{} \sqrt{u} ( 1 + u^2) du = \int\limits_{}^{} u^{0.5} + u^{2.5} du \]

\[ \frac{ u^{1.5} }{ 1.5 } + \frac{ u^{3.5} }{ 3.5 } + C\]

\[\frac{2 \sqrt{\tan^3 \theta} }{ 3 } + \frac{2 \sqrt{\tan^7 \theta} }{ 7 } + C\]

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