# How do you integrate int e^xsin4xdx using integration by parts?

Nov 3, 2015

$\int {e}^{x} \sin \left(4 x\right) \mathrm{dx} = \frac{{e}^{x} \sin \left(4 x\right) - 4 {e}^{x} \cos \left(4 x\right)}{17} + c$,
where $c$ is the integration constant.

#### Explanation:

$\int {e}^{x} \sin \left(4 x\right) \mathrm{dx} = - \frac{1}{4} \int {e}^{x} \frac{d}{\mathrm{dx}} \left(\cos \left(4 x\right)\right) \mathrm{dx}$

$= - \frac{1}{4} \left[{e}^{x} \cos \left(4 x\right) - \int \frac{d}{\mathrm{dx}} \left({e}^{x}\right) \cos \left(4 x\right) \mathrm{dx}\right]$

$= \frac{1}{4} \int {e}^{x} \cos \left(4 x\right) \mathrm{dx} - \frac{1}{4} {e}^{x} \cos \left(4 x\right)$

$= \frac{1}{16} \int {e}^{x} \frac{d}{\mathrm{dx}} \left(\sin \left(4 x\right)\right) \mathrm{dx} - \frac{1}{4} {e}^{x} \cos \left(4 x\right)$

$= \frac{1}{16} \left[{e}^{x} \sin \left(4 x\right) - \int \frac{d}{\mathrm{dx}} \left({e}^{x}\right) \sin \left(4 x\right) \mathrm{dx}\right] - \frac{1}{4} {e}^{x} \cos \left(4 x\right)$

$= \frac{1}{16} {e}^{x} \sin \left(4 x\right) - \frac{1}{16} \int {e}^{x} \sin \left(4 x\right) \mathrm{dx} - \frac{1}{4} {e}^{x} \cos \left(4 x\right)$

$\frac{17}{16} \int {e}^{x} \sin \left(4 x\right) \mathrm{dx} = \frac{1}{16} {e}^{x} \sin \left(4 x\right) - \frac{1}{4} {e}^{x} \cos \left(4 x\right) + {c}_{1}$,
where ${c}_{1}$ is the constant of integration.

$\int {e}^{x} \sin \left(4 x\right) \mathrm{dx} = \frac{{e}^{x} \sin \left(4 x\right) - 4 {e}^{x} \cos \left(4 x\right)}{17} + {c}_{2}$,
where ${c}_{2} = \frac{16 {c}_{1}}{17}$.