Table of Integrals, Series, and Products, 8th Edition

2.66 Combinations of trigonometric functions and exponentials

2.661 $\begin{array}{ccc}{\displaystyle \int e^{ax}{\text{sin}}^{p}x{\text{cos}}^{q}x\text{d}x}\hfill & =\hfill & \frac{1}{a^2+{\left(p+q\right)}^2}\{e^{ax}{\text{sin}}^{p}x{\text{cos}}^{q-1}x\left[a\text{cos}x+\left(p+q\right)\text{sin}x\right]\hfill \\ -\hfill & pa{\displaystyle \int e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q-1}x\text{d}x+\left(q-1\right)\left(p+q\right){\displaystyle \int e^{ax}{\text{sin}}^{p}x{\text{cos}}^{q-2}x\text{d}x}}\}\hfill \\ =\hfill & \frac{1}{a^2+{\left(p+q\right)}^2}\{e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q}x\left[a\text{sin}x-\left(p+q\right)\text{cos}x\right]\hfill \\ +\hfill & qa{\displaystyle \int e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q-1}x\text{d}x+\left(p-1\right)\left(p+q\right){\displaystyle \int e^{ax}{\text{sin}}^{p-2}x{\text{cos}}^{q}x\text{d}x}}\}\hfill \\ =\hfill & \frac{1}{a^2+{\left(p+q\right)}^2}\{e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q-1}x\left[a\text{sin}x\text{cos}x+q{\text{sin}}^{2}x-p{\text{cos}}^{2}x\right]\hfill \\ +\hfill & q\left(q-1\right){\displaystyle \int e^{ax}{\text{sin}}^{p}x{\text{cos}}^{q-2}x\text{d}x+p\left(p-1\right){\displaystyle \int e^{ax}{\text{sin}}^{p-2}x{\text{cos}}^{q}x\text{d}x}}\}\hfill \\ =\hfill & \frac{1}{a^2+{\left(p+q\right)}^2}\{e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q-1}x\left(a\text{sin}x\text{cos}x+q{\text{sin}}^{2}x-p{\text{cos}}^{2}x\right)\hfill \\ +\hfill & q\left(q-1\right){\displaystyle \int e^{ax}{\text{sin}}^{p-2}x{\text{cos}}^{q-2}x\text{d}x}\hfill \\ -\hfill & \left(q-p\right)\left(p+q-1\right){\displaystyle \int e^{ax}{\text{sin}}^{p-2}x{\text{cos}}^{q}x\text{d}x}\}\hfill \\ =\hfill & \frac{1}{a^2+{\left(p+q\right)}^2}\{e^{ax}{\text{sin}}^{p-1}x{\text{cos}}^{q-1}x\left(a\text{sin}x\text{cos}x+q{\text{sin}}^{2}x-p{\text{cos}}^{2}x\right)\hfill \\ +\hfill & p\left(p-1\right){\displaystyle \int e^{ax}{\text{sin}}^{p-2}x{\text{cos ...}}^{}}\hfill \end{array}$