How do you integrate int x^3cos(x^2) using integration by parts?

Aug 28, 2016

$= \frac{1}{2} {x}^{2} \sin \left({x}^{2}\right) + \frac{1}{2} \cos \left({x}^{2}\right) + C$

Explanation:

$\int {x}^{3} \cos \left({x}^{2}\right)$

$\int {x}^{2} . x \cos \left({x}^{2}\right)$

$= \int {x}^{2} \frac{d}{\mathrm{dx}} \left(\frac{1}{2} \sin \left({x}^{2}\right)\right)$

$= \frac{1}{2} {x}^{2} \sin \left({x}^{2}\right) - \frac{1}{2} \int \frac{d}{\mathrm{dx}} \left({x}^{2}\right) \sin \left({x}^{2}\right)$

$= \frac{1}{2} {x}^{2} \sin \left({x}^{2}\right) - \int x \sin \left({x}^{2}\right)$

$= \frac{1}{2} {x}^{2} \sin \left({x}^{2}\right) - \int \frac{d}{\mathrm{dx}} \left(- \frac{1}{2} \cos \left({x}^{2}\right)\right)$

$= \frac{1}{2} {x}^{2} \sin \left({x}^{2}\right) + \frac{1}{2} \cos \left({x}^{2}\right) + C$