Chapter 10.3, Problem 25E

25. Find the functions represented by the series obtained by the termwise integration of the given series from $-\pi$ to $x$.

a. $2{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^{n+1}}{n}\sin nx\sim x,-\pi <x<\pi }$

$\frac{4}{\pi }{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{\sin \left(2n+1\right)x}{\left(2n+1\right)}\sim f\left(x\right),}$

b. $f\left(x\right)=\{\begin{array}{l}-1,-\pi <x<0,\\ 1,0<x<\pi \end{array}$