A question about integration of a orthogonal function

Question

I have trouble figure out this orthogonal integration: $$ \Psi _{1}=sin(\frac{n\pi x}{a}),\Psi _{2}=cos(\frac{n\pi x}{a}) $$ $$ \int_{-\infty }^{+\infty}sin(\frac{n\pi x}{a})cos(\frac{n\pi x}{a})dx $$ $$\Psi _{1}$$ and $$\Psi _{2}$$ are orthogonals. This problem is not very complicated, I just need to prove $$ \int_{-\infty }^{+\infty}sin(\frac{n\pi x}{a})cos(\frac{n\pi x}{a})dx=\frac{1}{2}\int_{-\infty }^{+\infty}sin\frac{2n\pi x}{a}dx=0 $$ Here is why the problem bothers me. If the statement is a true statement, that $$ \int_{-\infty}^{+\infty}sinkxdx=0 $$ has to be true.However, the improper integral, $$ \int_{-\infty}^{+\infty}sinkxdx $$ is divergent. This is contradicted to the statement. So how should I understand this. And we know the Cauchy principal which shows that $$ \int_{-\infty}^{+\infty}sinkxdx=\lim_{A \to +\infty}\int_{-A}^{+A}sinkxdx=0 $$ So how can I know when I can use the Cauchy principal to solve problems? Or is this principal helpful to understand this question?