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

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Theorem

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

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


Then:

$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$


That is:

$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \pi \delta_{m n}$

where $\delta_{m n}$ is the Kronecker delta.


Proof

Let $m \ne n$.

\(\ds \int \sin m x \sin n x \rd x\) \(=\) \(\ds \frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C\) Primitive of $\sin m x \sin n x$
\(\ds \leadsto \ \ \) \(\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x\) \(=\) \(\ds \intlimits {\frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } } \alpha {\alpha + 2 \pi}\)
\(\ds \) \(=\) \(\ds \paren {\frac {\sin \paren {\paren {m - n} \paren {\alpha + 2 \pi} } } {2 \paren {m - n} } - \frac {\sin \paren {\paren {m + n} \paren {\alpha + 2 \pi} } } {2 \paren {m + n} } }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\frac {\sin \paren {m - n} \alpha} {2 \paren {m - n} } - \frac {\sin \paren {m + n} \alpha} {2 \paren {m + n} } }\)
\(\ds \) \(=\) \(\ds \paren {\frac {\sin \paren {m - n} \alpha} {2 \paren {m - n} } - \frac {\sin \paren {m + n} \alpha} {2 \paren {m + n} } }\) Sine of Angle plus Full Angle: Corollary
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\frac {\sin \paren {m - n} \alpha} {2 \paren {m - n} } - \frac {\sin \paren {m + n} \alpha} {2 \paren {m + n} } }\)
\(\ds \) \(=\) \(\ds 0\)

$\Box$


When $m = n$ we have:

\(\ds \int \sin m x \sin m x \rd x\) \(=\) \(\ds \int \sin^2 m x \rd x\)
\(\ds \) \(=\) \(\ds \frac x 2 - \frac {\sin 2 m x} {4 m} + C\) Primitive of $\sin^2 m x$
\(\ds \leadsto \ \ \) \(\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin m x \rd x\) \(=\) \(\ds \intlimits {\frac x 2 - \frac {\sin 2 m x} {4 m} } \alpha {\alpha + 2 \pi}\)
\(\ds \) \(=\) \(\ds \paren {\frac {\alpha + 2 \pi} 2 - \frac {\sin \paren {2 m \paren {\alpha + 2 \pi} } } {4 m} } - \paren {\frac \alpha 2 - \frac {\sin 2 m \alpha} {4 m} }\)
\(\ds \) \(=\) \(\ds \paren {\frac {\alpha + 2 \pi} 2 - \frac {\sin 2 m \alpha} {4 m} } - \paren {\frac \alpha 2 - \frac {\sin 2 m \alpha} {4 m} }\) Sine of Angle plus Full Angle: Corollary
\(\ds \) \(=\) \(\ds \frac {\alpha + 2 \pi - \alpha} 2\)
\(\ds \) \(=\) \(\ds \pi\)

$\blacksquare$


Sources

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