Chapter 8.2, Problem 73E

Two useful exponential integrals Use integration by parts to derive the following formulas for real numbers a and b.

$\begin{array}{l}{\displaystyle \int e^{ax}\sin bxdx}=\frac{e^{ax}(a\sin bx-b\cos bx)}{a^2+b^2}+C\\ {\displaystyle \int e^{ax}\cos bxdx}=\frac{e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}+C\end{array}$