Chapter 7.2, Problem 76ES

(a) In the integral ${\displaystyle \int x\cos xdx,}$ let

$\begin{array}{l}u=x,dv=\cos xdx,\\ du=dx,v=\sin x+C_1\end{array}$

Show that the constant $C_1$ cancels out, thus giving the same solution obtained by omitting $C_1.$

(b) show that in general

$uv-{\displaystyle \int v}du=u(v+C_1)-{\displaystyle \int (v+}{C}_1)du$

thereby justifying the omission of the constant of integration when v in integration by parts.