# isn&#39;t this wikipedia Definite integrals involving rational or irrational expression wrong!?

isn't this wikipedia Definite integrals involving rational or irrational expression wrong!?

According to my calculations: $$\sin[\frac{(m+1)\pi}{n}]$$ while in wiki it is $$\sin[\frac{(m+1)}{n}]$$

seems $$\pi$$ missed.

here is result of my work:

$$\int_0^\infty \frac{x^m \, dx}{({x^n+a^n)}^r}=\frac{(-1)^{r-1}\pi a^{m+1-nr}\Gamma [(m+1)/n]}{n\sin[\pi (m+1)/n](r-1)!\Gamma[(m+1)/n-r+1]} \ \ , 0

Remark: @icurays1 in your link there is another wrong $$\Gamma [\frac {(m+1)}{(n-p+1)}]$$ while it could be $$\Gamma [(\frac {m+1}{n})-p+1]$$

• @icurays1 i check several times it seems to me $\pi$ is missed. – Neo Jan 30 '13 at 15:37
• What would Jesus Wolfram Alpha say? – nbubis Jan 30 '13 at 15:39
• I've deleted my previous comment - @neo you might be correct, I've found this which agrees with you. Someone should verify with a published table though, for instance Gradshteyn-Ryzhik – icurays1 Jan 30 '13 at 15:56
• Well, something is wrong on that Wikipedia page. The second expression is a special case of the one you listed above, but the right hand sides do not agree. – Willie Wong Jan 30 '13 at 16:10
• @nbubis Most of the time Jesus Wolfram Alpha says "calculation too complicated...give me your money and I will do it!" ;-) – Matemáticos Chibchas Jan 30 '13 at 16:25

It is definitely wrong. Take $a=1, m=0, n=2, r=1$. Then it reads $$\int_0^\infty \frac{1}{x^2+1}\,dx = \frac{\pi}{2 \sin(1/2)}$$ which is false. The correct value is $\pi/2$, which agrees with your proposed $\sin\left[\frac{(m+1)\pi}{n}\right]$.