# How do you use Integration by Substitution to find intx*sin(x^2)dx?

Sep 26, 2014

Begin by letting making a u-substitution.

Let $u = {x}^{2}$

$\mathrm{du} = 2 x \mathrm{dx}$

$\frac{\mathrm{du}}{2} = x \mathrm{dx} \implies$ This can be substituted in the original integral

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

$= \frac{1}{2} \left[- \cos \left(u\right)\right] + C$

$= - \frac{1}{2} \left[\cos \left(u\right)\right] + C$

Rewrite in terms of $x$

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