# How do you integrate int sin(sqrtx) by integration by parts method?

Sep 2, 2016

$2 \sin \left(\sqrt{x}\right) - 2 \sqrt{x} \cos \left(\sqrt{x}\right) + C$

#### Explanation:

Making $x = {y}^{2}$ in $\int \sin \left(\sqrt{x}\right) \mathrm{dx}$

after $\mathrm{dx} = 2 y \mathrm{dy}$ we have

$\int \sin \left(\sqrt{x}\right) \mathrm{dx} \equiv 2 \int y \sin y \mathrm{dy}$

but $\frac{d}{\mathrm{dy}} \left(y \cos y\right) = \cos y - y \sin y$ so

$2 \int y \sin y \mathrm{dy} = 2 \int \cos y \mathrm{dy} - 2 y \cos y = 2 \sin y - 2 y \cos y + C$

Finally

$\int \sin \left(\sqrt{x}\right) \mathrm{dx} = 2 \sin \left(\sqrt{x}\right) - 2 \sqrt{x} \cos \left(\sqrt{x}\right) + C$