1. ## Integrating sin(sqrt x)?

I need to solve the problem:

$\int\sin\sqrt x\ dx$

Using integration by parts. So far, I've done:

$u=\sqrt x$
$du=\frac{1}{2\sqrt x}\ dx$
$2udu=dx$

But then I'm lost.

2. Hello, quarks!

You're off to a good start . . .

$\int\sin(\sqrt{x})\,dx$

So far, I've done:

$w\:=\:\sqrt x \quad\Rightarrow x \:=\:w^2\quad\Rightarrow\quad dx \:=\:2w\,dw$

Substitute: . $\int 2w\sin w\,dw$

Integrate by parts:
. . $\begin{array}{ccccccc}u &=&2w & &dv &=&\sin w\,dw \\ du &=&2\,dw & & v &=&-\cos w\end{array}$

And we have: . $-2w\cos w + 2\int\cos w\,dw \;=\;-2w\cos w + 2\sin w + C$

Back-substitute: . $-2\sqrt{x}\cos(\sqrt{x}) + 2\sin(\sqrt{x}) + C$