∫ sin(√x) dx = ?

Let $u = \sqrt{x}$.
Then $u^2 = x$.
Differentiating both sides with respect to $u$ gives $2u\,du = dx$.

Substitute $u$ and $dx$ into the integral:

For the integral $\int u \sin(u)\,du$, use the integration by parts formula $\int v\,dw = vw - \int w\,dv$.
Let $v = u$ and $dw = \sin(u)\,du$.
Then $dv = du$ and $w = \int \sin(u)\,du = -\cos(u)$.

Now, substitute this back into the expression from Step 2:

Replace $u$ with $\sqrt{x}$ to express the answer in terms of $x$:

The final answer is $2\sin(\sqrt{x}) - 2\sqrt{x}\cos(\sqrt{x}) + C$.