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:

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

Proof

Let $m \ne n$.

\(\ds \int \sin m x \cos n x \rd x\)

\(=\)

\(\ds \frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } + C\)

Primitive of $\sin m x \cos n x$

\(\ds \leadsto \ \ \)

\(\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x\)

\(=\)

\(\ds \intlimits {\frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } } \alpha {\alpha + 2 \pi}\)

\(\ds \)

\(=\)

\(\ds \paren {\frac {-\map \cos {\paren {m - n} \paren {\alpha + 2 \pi} } } {2 \paren {m - n} } - \frac {\map \cos {\paren {m + n} \paren {\alpha + 2 \pi} } } {2 \paren {m + n} } }\)

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }\)

\(\ds \)

\(=\)

\(\ds \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }\)

Cosine of Angle plus Full Angle: Corollary

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \paren {\frac {-\map \cos {m - n} \alpha} {2 \paren {m - n} } - \frac {\map \cos {m + n} \alpha} {2 \paren {m + n} } }\)

\(\ds \)

\(=\)

\(\ds 0\)

after simplification

When $m = n$ we have:

\(\ds \int \sin m x \cos m x \rd x\)

\(=\)

\(\ds \frac {\sin^2 m x} {2 m} + C\)

Primitive of $\sin m x \cos m x$

\(\ds \leadsto \ \ \)

\(\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \cos m x \rd x\)

\(=\)

\(\ds \intlimits {\frac {\sin^2 m x} {2 m} } \alpha {\alpha + 2 \pi}\)

\(\ds \)

\(=\)

\(\ds \frac {\map {\sin^2} {m \paren {\alpha + 2 \pi} } } {2 m} - \frac {\sin^2 m \alpha} {2 m}\)

\(\ds \)

\(=\)

\(\ds \frac {\sin^2 m \alpha} {2 m} - \frac {\sin^2 m \alpha} {2 m}\)

Cosine of Angle plus Full Angle: Corollary

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 2$. Fourier Series: $(3)$