Answer
$y=\sqrt x-cos(\pi x)$
Work Step by Step
Step 1. Given $\frac{dy}{dx}=\frac{1}{2\sqrt x}+\pi\ sin(\pi x)$, we have $y=\int(\frac{1}{2\sqrt x}+\pi\ sin(\pi x))dx=\sqrt x-cos(\pi x)+C$ where $C$ is a constant.
Step 2. Using the point on the curve, $y(1)=2$, we have$\sqrt 1-cos(\pi)+C=2$ and $C=0$.
Step 3. Thus the answer is $y=\sqrt x-cos(\pi x)$
