<s>[INST] How to prove $\int^{\infty}_0 e^{-c^2/a^2}c^4\,dc=\frac{3}{8}a^5\sqrt\pi$? How do i prove this integral can someone give me proof i've been trying this too long i tried direct integration by parts, differentiation all to no avail.Please support us :)

$$\int^{\infty}_0 e^{\frac{-c^2}{a^2}}c^4\,dc=\frac{3}{8}a^5\sqrt\pi$$

 [/INST] Two times by parts:
$\int\limits_0^\infty e^\frac{-x^2}{a^2}dx=
\left.xe^\frac{-x^2}{a^2}\right|_0^\infty+
\int\limits_0^\infty \frac{2x^2}{a^2}e^\frac{-x^2}{a^2}dx=
\frac{2}{a^2}\int\limits_0^\infty e^\frac{-x^2}{a^2}d\left(\frac{x^3}{3}\right)=$
$
\frac{2}{a^2}\left(\left.\frac{x^3}{3}e^\frac{-x^2}{a^2}\right|_0^\infty+\int\limits_0^\infty \frac{x^3}{3}\cdot\frac{2x}{a^2}e^\frac{-x^2}{a^2}dx\right)=
\frac{4}{3a^4}\int\limits_0^\infty x^4e^\frac{-x^2}{a^2}dx$
Let alone $\int\limits_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$ or here be a known result, we obtain
$$\int\limits_0^\infty x^4e^\frac{-x^2}{a^2}dx=\frac{3a^4}{4}\int\limits_0^\infty e^\frac{-x^2}{a^2}dx=
\frac{3a^4}{4}\cdot a\int\limits_0^\infty e^{-x^2}dx=\frac{3a^5\sqrt{\pi}}{8}\hbox{, QED}$$
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