Please show me the steps of the following integration. I got an answer in Wolfram, but I need steps..

$$\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$$

  • $\begingroup$ @draks here it is. And WA do not know closed form for this integral $\endgroup$ – Norbert Jul 31 '12 at 11:02
  • $\begingroup$ Well, I'm sorry its not arctan, its just tan $\endgroup$ – Anubis Jul 31 '12 at 11:05
  • $\begingroup$ This is the link $\endgroup$ – Anubis Jul 31 '12 at 11:08
  • $\begingroup$ @draks sorry, I've typed it wrong. It's now corrected. Can you evaluate now? $\endgroup$ – Anubis Jul 31 '12 at 11:10
  • $\begingroup$ Aha, so please edit your question accordingly!!! and click on "show steps" in your linked W|A page... $\endgroup$ – draks ... Jul 31 '12 at 11:10

We try the substitution $t^3 = \tan^2 x$. Therefore, $3t^2 dt = 2 \tan x \sec^2 x dx$, giving us $\frac{dx}{\sqrt{t}} = \frac{3 dt}{2(1+t^3)}$.

Thus, we will only evaluate $\int \frac{3 dt}{1+t^3} $, divide by $2$ and substitute back. Note that $3 = (1-t+t^2) + (2-t)(1+t)$, reducing our integral to $$ \int \frac{dt}{1+t} + \int \frac{(2-t)dt}{1-t + t^2} $$ I won't elaborate further, since our integrals are already in standard forms.

  • $\begingroup$ Wow..a cleaver approach. You are a genius..! $\endgroup$ – Anubis Aug 1 '12 at 9:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.