# Integral over 2 pi of Sine of m x by Cosine of n x

## Theorem

Let $m, n \in \Z$ be integers.

Let $\alpha \in \R$ be a real number.

Then:

$\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$

## Proof

Let $m \ne n$.

 $\displaystyle \int \sin m x \cos n x \rd x$ $=$ $\displaystyle \frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } + C$ Primitive of $\sin m x \cos n x$ $\displaystyle \leadsto \ \$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x$ $=$ $\displaystyle \intlimits {\frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } } \alpha {\alpha + 2 \pi}$ $\displaystyle$ $=$ $\displaystyle \paren {\frac {-\map \cos {\paren {m - n} \paren {\alpha + 2 \pi} } } {2 \paren {m - n} } - \frac {\map \cos {\paren {m + n} \paren {\alpha + 2 \pi} } } {2 \paren {m + n} } }$ $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }$ $\displaystyle$ $=$ $\displaystyle \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }$ Corollary to Cosine of Angle plus Full Angle $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }$ $\displaystyle$ $=$ $\displaystyle 0$ after simplification

When $m = n$ we have:

 $\displaystyle \int \sin m x \cos m x \rd x$ $=$ $\displaystyle \frac {\sin^2 m x} {2 m} + C$ Primitive of $\sin m x \cos m x$ $\displaystyle \leadsto \ \$ $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos m x \rd x$ $=$ $\displaystyle \intlimits {\frac {\sin^2 m x} {2 m} } \alpha {\alpha + 2 \pi}$ $\displaystyle$ $=$ $\displaystyle \frac {\map {\sin^2} {m \paren {\alpha + 2 \pi} } } {2 m} - \frac {\sin^2 m \alpha} {2 m}$ $\displaystyle$ $=$ $\displaystyle \frac {\sin^2 m \alpha} {2 m} - \frac {\sin^2 m \alpha} {2 m}$ Corollary to Cosine of Angle plus Full Angle $\displaystyle$ $=$ $\displaystyle 0$

$\blacksquare$