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

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.

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

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  • $\begingroup$ 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. $\endgroup$ – user223391 Nov 1 '15 at 22:02
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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.

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

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