Evaluating the indefinite integral $\int \tan \sqrt {x} \,dx$

Question

$$\int \tan \sqrt {x} \,dx$$

I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly.

Answer 1

This is one of the trig functions that cannot be integrated in

the usual way. I suggest using Reimann sums to approximate or other methods.

Answer 2

I fiddled with this one a little bit: \begin{align} u & = \sqrt x \\[10pt] u^2 & = x \\[10pt] 2u\,du & = dx \\[10pt] \int \tan \sqrt x \, dx & = 2 \int u\ \ \underbrace{\tan u\ du}_{dv} = \underbrace{2\int u\,dv = 2uv - 2\int v\,du}_\text{integration by parts with $dv$ as below}. \tag 1 \\[20pt] dv & = \tan u \, du \\ v & = -\log|\cos u| \\[10pt] \text{So the expression in $(1)$ is } & -2u \log |\cos u| + 2\int \log |\cos u| \, du. \end{align}

Then I resorted to Wolfram.