Why is it equal?

Question

Which properties of integral and cos are used here?

$$\frac 2l \int_0^l x\cos\frac{k\pi x} l\,dx=\frac{2l}{\pi^2}\int_0^\pi x\cos kx \,dx$$

where x from l to l.

Answer 1

$$\frac 2l \int_0^l x\cos\frac{k\pi x} l\,dx=\frac{2l}{\pi^2}\int_0^\pi x\cos kx \,dx$$ This is a substitution, of the form $u=\frac{\pi x}l$.

Substitution into the first integral gives $$\frac2l\int_0^\pi\frac l\pi u\cos ku\cdot \frac{l}{u}\,du=\frac{2l}{\pi^2}\int_0^l u\cos ku\,du=\frac{2l}{\pi^2}\int_0^l x\cos kx\,dx$$ where the last step is just relabelling $u$ by $x$ since they are both dummy variables.