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

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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 $m \ne n$.

\(\displaystyle \int \sin m x \cos n x \rd x\) \(=\) \(\displaystyle \frac {-\cos \left({m - n}\right) x} {2 \left({m - n}\right)} - \frac {\cos \left({m + n}\right) x} {2 \left({m + n}\right)} + C\) Primitive of $\sin m x \cos n x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x\) \(=\) \(\displaystyle \left[{\frac {-\cos \left({m - n}\right) x} {2 \left({m - n}\right)} - \frac {\cos \left({m + n}\right) x} {2 \left({m + n}\right)} }\right]_\alpha^{\alpha + 2 \pi}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {-\cos \left({\left({m - n}\right) \left({\alpha + 2 \pi}\right)}\right)} {2 \left({m - n}\right)} - \frac {\cos \left({\left({m + n}\right) \left({\alpha + 2 \pi}\right)}\right)} {2 \left({m + n}\right)} }\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \left({\frac {-\cos \left({m - n}\right) \alpha} {2 \left({m - n}\right)} - \frac {\cos \left({m + n}\right) \alpha} {2 \left({m + n}\right)} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {-\cos \left({m - n}\right) \alpha} {2 \left({m - n}\right)} - \frac {\cos \left({m + n}\right) \alpha} {2 \left({m + n}\right)} }\right)\) Corollary to Cosine of Angle plus Full Angle
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \left({\frac {-\cos \left({m - n}\right) \alpha} {2 \left({m - n}\right)} - \frac {\cos \left({m + n}\right) \alpha} {2 \left({m + n}\right)} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 0\) 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 $\sin m x \cos m x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos m x \rd x\) \(=\) \(\displaystyle \left[{\frac {\sin^2 m x} {2 m} }\right]_\alpha^{\alpha + 2 \pi}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin^2 \left({m \left({\alpha + 2 \pi}\right)}\right)} {2 m} - \frac {\sin^2 m \alpha} {2 m}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin^2 m \alpha} {2 m} - \frac {\sin^2 m \alpha} {2 m}\) Corollary to Cosine of Angle plus Full Angle
\(\displaystyle \) \(=\) \(\displaystyle 0\)

$\blacksquare$


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