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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..

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

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  • $\begingroup$ Thabkyou ! I had a feeling it would be something stupidly simple! Thanks! $\endgroup$ – Ellipse Jan 17 at 12:29

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