Integral over 2 pi of Cosine of n x

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Theorem

Let $m \in \Z$ be an integer.


Then:

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


Proof

Let $m \ne 0$.

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

$\Box$


Let $m = 0$.

\(\ds \int \cos 0 x \rd x\) \(=\) \(\ds \int 1 \rd x\) Cosine of Zero is One
\(\ds \) \(=\) \(\ds x + C\) Primitive of Constant
\(\ds \leadsto \ \ \) \(\ds \int_\alpha^{\alpha + 2 \pi} \cos 0 x \rd x\) \(=\) \(\ds \bigintlimits x \alpha {\alpha + 2 \pi}\)
\(\ds \) \(=\) \(\ds \alpha + 2 \pi - \alpha\)
\(\ds \) \(=\) \(\ds 2 \pi\)

$\blacksquare$


Sources

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