Mixed sine and cosine

Integrating by parts,

$$L := \int_0^\pi \sin mx \cos nx \d x$$

$$= \frac{1}{n} \biggl[ \sin mx \sin nx \biggr]_0^\pi - \frac{m}{n} \int_0^\pi \cos mx \sin nx \d x$$

$$= - \frac{m}{n} \biggl[ \frac{1}{m-n} \biggl( (-1)^{n-m} - 1 \biggr) + L \biggr]$$

(by (3)). Hence,

$$L \biggl( 1 + \frac{m}{n} \biggr) = \frac{m}{n(n-m)} \biggl( (-1)^{n-m} - 1 \biggr)$$

$$L = \frac{\frac{m}{n-m} \biggl( (-1)^{n-m} - 1 \biggr)}{n+m}$$

$$= \frac{m}{n^2 - m^2} \biggl( (-1)^{n-m} - 1 \biggr)$$

When $n-m$ is odd, $(-1)^{n-m} = -1$ and when $n-m$ is even, $(-1)^{n-m} = 1$. Thus, we can express this as

$$\int_0^\pi \sin mx \cos nx \d x = \begin{cases} 0, & n-m \text{ even} \\ 2m/(m^2 - n^2), & n-m \text{ odd} \end{cases}$$