# Integrate by parts ? 15x sinx dx

Apr 10, 2018

$- 15 x \cos \left(x\right) + 15 \sin \left(x\right) + c$

#### Explanation:

$\int \left(15 x \sin \left(x\right)\right) \mathrm{dx}$

Since $\sin \left(x\right)$ is a periodic function, it is going to be $u '$

$u ' = \sin \left(x\right) \text{ } u = - \cos \left(x\right)$
$v = 15 x \text{ } v ' = 15$

$\int \left(v \cdot u '\right) = v \cdot u - \int \left(v ' \cdot u\right)$

$\int \left(15 x \sin \left(x\right)\right) \mathrm{dx} = - 15 x \cos \left(x\right) - \int \left(- 15 \cos \left(x\right)\right) \mathrm{dx}$
$- 15 x \cos \left(x\right) + \int \left(15 \cos \left(x\right)\right) \mathrm{dx} = - 15 x \cos \left(x\right) + 15 \sin \left(x\right) + c$