# Integral $\int_{0}^{D} x \cdot (\sqrt{1+x^a})^{b}\,dx$

do you have any ideas on solving the following integrals: $$\int_{0}^{D} x \cdot (\sqrt{1+x^a})^{b}\,dx$$ where $b$ are positive integers and a is also positive.

For the special case of $a=2$, it is straightforward to solve; however for more general values, it is really tricky and change of RV using $\sqrt{1+x^a} = y$ seems not to make the integral easier.

Base on the requirements to solve binomial integral using elementary functions, there should be no solution to the integral using elementary functions; but how about using hypergeometric functions?

Thanks.

## 1 Answer

This is an example of a binomial integral. It can be rewritten as $$\int_{0}^{D}x^1\left(1+x^a\right)^{b/2}\,\mathrm{d}x$$ See the link for how to solve it

• Hi, @Teh, thanks, but to solve the binomial integrals with elementary functions, one of the three requirements needs to be met in our case: 1) $\frac b2$ is an integer, 2) $\frac2a$ is an integer, 3) $\frac2a+\frac b2$ is an integer; however these three requirements does not necessarily hold for our case. So it can be confirmed at least that there is no solution to the integral using elementary functions. – Vic Jan 19 '18 at 18:50
• @Vic exactly, so if it does not meet any of those cases, it cannot be done and that is the “general” solution – Teh Rod Jan 19 '18 at 18:52
• hi, @Teh, I am thinking whether it is possible to solve it use hypergeometric functions such as Meijer G function or others. – Vic Jan 19 '18 at 18:56