Partial integration and substitution

Question

I have trouble solving

$$\int_0^\infty \frac{x^2}{\lambda^2}\cdot e^{-(x/\lambda)^2/2}\,dx$$

Somehow I end up with a '2' too much. The correct result is lambda * sqrt(pi/2).

Does anyone know how to solve this?

Answer 1

You have an extra $2$ in your parts integration. Let's see:

\begin{align} \int_0^\infty \frac{x^2}{\lambda^2} e^{-\frac{(x/\lambda)^2}{2}} dx &= -\int_0^\infty x \frac{d}{d x} e^{-\frac{(x/\lambda)^2}{2}} dx \\ \\ &= \int_0^\infty e^{-\frac{(x/\lambda)^2}{2}} dx \\ \\ &= \sqrt{2}\lambda\int_0^\infty e^{-\frac{(x/\lambda)^2}{2}} d\left(\frac{x}{\sqrt{2}\lambda}\right) = \sqrt{\frac{\pi}{2}}\lambda \end{align}