Answer
$\psi_{(x)} = (\frac{2}{L})^{1/2} sin(\frac{2\pi x}{L})$
Work Step by Step
Given $\psi_{(x)} = sin(\frac{2\pi x}{L})$ for $0 \le x\le L$
$$N^2\int_0^L sin^2(\frac{2\pi x}{L}) dx = 1$$
$$N^2\times \frac{1}{2}\int_0^L \big[1 - cos(\frac{4\pi x}{L})\big] dx = 1$$
$$N^2 \times \frac{1}{2}\big[x - \frac{sin(\frac{4\pi x}{L})}{\frac{4\pi}{L}}\big]_0^L dx = 1$$
$$N^2 \frac{L}{2} = 1$$
$$ N = (\frac{2}{L})^{1/2}$$
Hence the normalized wave function is
$$ (\frac{2}{L})^{1/2} sin(\frac{2\pi x}{L})$$
