Question

Find the integral:

$\int \cos \sqrt{x}dx.$

Answer 1

$\int \cos \sqrt{x}dx.$
$x=t^2$
$dx=2tdx$
$=\int \mathrm{cost}2tdx$
$=2\int t.\cos tdx$
$=2\left[t.\int \cos tdx-\int \frac{d}{dt}\left(t\right).\int \cos tdt)dx\right]$
$=2\left[t.\sin t-\int \sin t-dt\right]$
$=2\left[t.\sin t+\cos t\right]+c$
$=2\left[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}\right]+c$