Answer
The solution is $$\int\frac{sin\sqrt\theta}{\sqrt\theta} d\theta=-2\cos \sqrt\theta+c.$$
Work Step by Step
To solve the integral $$\int\frac{sin\sqrt\theta}{\sqrt\theta} d\theta$$ we will use substitution $t=\sqrt\theta\Rightarrow dt=\frac{d\theta}{2\sqrt\theta}\Rightarrow 2dt=\frac{d\theta}{\sqrt\theta}$ and putting this in the integral we have:
$$\int\frac{sin\sqrt\theta}{\sqrt\theta} d\theta=\int 2\sin tdt=-2\cos t+c,$$
where $c$ is arbitrary constant. Now we have to express our solution (which is in terms of $t$) in terms of $\theta$:
$$\int\frac{sin\sqrt\theta}{\sqrt\theta} d\theta=-2\cos t+c=-2\cos \sqrt\theta+c.$$
