1
$\begingroup$

How do I calculate the integral

$$I=\int\limits_0^1\frac{1}{1+x^\frac{\pi}{2}}dx$$

I have no idea about the gamma and beta function. In the comments it has been proposed to use gamma and beta functions. how can I go about doing it especially the use of gamma and beta functions.

$\endgroup$

closed as off-topic by Jack D'Aurizio Jun 15 '18 at 16:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jack D'Aurizio
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Do you mean $$\int\frac{1}{(1+x)^{\pi/2}}dx$$? $\endgroup$ – Dr. Sonnhard Graubner Jun 15 '18 at 16:44
  • $\begingroup$ Are you able to solve $\int \frac {dx}{x^{\alpha}}$? $\endgroup$ – lulu Jun 15 '18 at 16:45
  • 2
    $\begingroup$ Use $x=\tan^{4/\pi}t$ and some basic facts about the Beta and Gamma functions. $\endgroup$ – J.G. Jun 19 '18 at 6:27
  • 1
    $\begingroup$ The closed form of this involves polygamma functions. It is highly unlikely that the average evaluator of integrals is going to know about them $\endgroup$ – graveolensa Aug 29 '18 at 14:26
  • 1
    $\begingroup$ $\int_{0}^{1} \frac{dx}{1+x^{\pi/2}}=\frac{\psi ^{(0)}\left(\frac{1}{2}+\frac{1}{\pi }\right)-\psi^{(0)}\left(\frac{1}{\pi }\right)}{\pi }$ where $\psi$ is the polygamma function. See my earlier question math.stackexchange.com/questions/42093/… $\endgroup$ – graveolensa Aug 30 '18 at 21:34