# Integral trick explanation or link

We used the following integral trick in our lecture, I really can't understand why it works. I would appreciate an explanation or even just a name so i can search for it :)

So apparently:

$$\int_{-\infty}^\infty\boldsymbol{x}\cdot f(x)dx=\int_{-\infty}^\infty\boldsymbol{(x-x_{a})}f(x)dx+\boldsymbol{x}_{\boldsymbol{a}}\cdot\int_{-\infty}^\infty f(x)dx$$

Where $$f(x)$$ is some function of $$x$$ and $$x_a$$ is .. not sure about that either, probably any constant..

• This is just linearity of the integral. – Nicolas Jan 17 at 12:26
• Is your choice of what to make bold and where to place $\cdot$ meant to imply certain quantities are vectors and hence so is $dx$? – J.G. Jan 17 at 12:28

This is just because $$xf(x)=(x-x_a)f(x)+x_af(x)$$ $$x_a$$ is a constant, so you can pull it outside of the integral.