3
$\begingroup$

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

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

  • 0
    Aha, so please edit yo$u$r question accordingly!!! and click on "show steps" in your linked W|A page...2012-07-31

1 Answers 1

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.

  • 0
    Wow..a cleaver approach. You are a genius..!2012-08-01