# Thread: Integrating x^3 * csc^3((pi*x^2)/2)

1. ## Integrating x^3 * csc^3((pi*x^2)/2)

I have been having a problem trying to calculate this integral:

$\displaystyle \int{\frac {{x}^{3}}{ \left( \sin \left( 1/2\,\pi\,{x}^{2} \right) \right) ^{3}}}$

and the function:
$\displaystyle \int{\frac {{x}^{5}}{ \left( \sin \left( 1/2\,\pi\,{x}^{2} \right) \right) ^{5}}}$

If someone could give me a start on it that would be amazing!

MT

2. They are unelementary integrals.
You can not solve them using elementary methods.
Did you post the whole question?

3. I'm not so sure they are not "elementary". They certanly can be simplified. In each case, let $\displaystyle u= \frac{\pi x^2}{2}$ so that $\displaystyle du= \pi x dx$ and $\displaystyle \frac{1}{\pi}du= xdx$. Also, $\displaystyle x^2= \frac{u}{\pi}$.

Now factor an "x" out of the numerator:
$\displaystyle \int \frac{x^3 dx}{sin((1/2)\pi x^2)}= \int \frac{(x^2)(xdx)}{sin((1/2)\pi x^2)}$$\displaystyle = \int\fra{(u/\pi)(\frac{1}{pi}du}{sin(u)}= \frac{1}{\pi^2}\int u csc(u)du$

Similarly, $\displaystyle \int \frac{x^5 dx}{sin((1/2)\pi x^2)}= \frac{1}{\pi^3}\int u^2 csc(u)du$

Try integration by parts now.

4. I did post the correct integral I need to solve.

HallsofIvy:

I think my whole issue is keeps coming down to solving:

$\displaystyle \int{u*csc{(u)} du}$

I tried to use integration by parts but in the last integrand I get a nasty equation for which I don't have the experience to integrate.