This are notes from a NN. This is one of many steps to solve differential equations with separate variables. In this step NN does following:
$$g_1(y) = \sum_{n=1}^\infty A_n \sin\left(\frac{n\cdot\pi}{b}y\right)$$
And then, to obtain $A_n$ he does:
$$A_n = \frac{2}{b}\int_0^b g_1(y)\cdot \sin\left(\frac{n\pi}{b}y\right)dx$$
Why? Or what is the name of this method?
Thx