Question

Evaluate the integral of cos(sqrt(x))dx.

Answer 1

Let us evaluate the integral $\int\cos(\sqrt x)dx$.

Let us use the transformation $t=\sqrt x.$ Then $x=t^2,$ and hence $dx=2tdt.$

It follows that

$\int\cos(\sqrt x)dx=2\int t\cos tdt$

$|u=t,dv=\cos tdt,\ du=dt, v=\sin t|$

$=2t\sin t-2\int \sin t dt=2t\sin t+2\cos t +C$

$=2\sqrt x\sin (\sqrt x)+2\cos (\sqrt x) +C.$