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

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

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.

• I don't believe there's a closed-form expression for this integral. – Travis Willse Nov 1 '15 at 15:18
• Are you sure that it's an indefinite integral? – user258700 Nov 1 '15 at 15:18
• Yeah. Indefinite. It took long time to solve. But i did it. I want to know a shoter method. – AVIE Nov 1 '15 at 15:23
• you will get the PolyLog function – Dr. Sonnhard Graubner Nov 1 '15 at 15:25
• I don't know what is polylog. Could you please explain that – AVIE Nov 1 '15 at 15:26

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.

• Choose your words carefully. The function is indeed integrable, in the sense that the Riemann sums converge. It's not expressible in terms of elementary functions however. – user223391 Nov 1 '15 at 22:02

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.

• Could you please explain the fourth line – AVIE Nov 2 '15 at 0:12
• @AVIE : The fourth line is integration by parts. ${}\qquad{}$ – Michael Hardy Nov 2 '15 at 0:50
• Thank you!! Sir. Appreciate your work – AVIE Nov 2 '15 at 15:47