Answer
$$\int\cos\sqrt xdx=2(\sqrt x\sin \sqrt x+\cos\sqrt x)+C$$
Work Step by Step
$$A=\int\cos\sqrt xdx$$
We set $a=\sqrt x$, which means $$da=\frac{1}{2\sqrt x}dx=\frac{1}{2a}dx$$ $$dx=2ada$$
Therefore, $$A=2\int a\cos ada$$
Next, we do integration by parts.
Take $u=a$ and $dv=\cos ada$
We would have $du=da$ and $v=\sin a$
Applying the formula $\int udv=uv-\int vdu$, we have
$$A=2(a\sin a-\int \sin ada)$$ $$A=2(a\sin a+\cos a)+C$$ $$A=2(\sqrt x\sin \sqrt x+\cos\sqrt x)+C$$
