Primitive of Power of x by Cosine of a x

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Theorem

$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x$


Corollary

$\ds \int x^m \cos a x \rd x = \sum_{k \mathop = 1}^{m + 1} \paren {m^{\underline {k - 1} } \frac {x^{m + 1 - k} } {a^k} \map {\sin} {x + \dfrac {\pi} 2 \paren {k - 1} } }$

where $m^{\underline {k - 1} }$ denotes the $k - 1$th falling factorial of $m$.


Proof

Lemma

$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a - \frac m a \int x^{m - 1} \sin a x \rd x$

$\Box$


From Primitive of $x^{m - 1} \sin a x$: Lemma:

$(1): \quad \ds \int x^{m - 1} \sin a x \rd x = \frac {-x^{m - 1} \cos a x} a + \frac {m - 1} a \int x^{m - 2} \cos a x \rd x$


So:

\(\ds \int x^m \cos a x \rd x\) \(=\) \(\ds \frac {x^m \sin a x} a - \frac m a \int x^{m - 1} \sin a x \rd x\) Lemma
\(\ds \) \(=\) \(\ds \frac {x^m \sin a x} a - \frac m a \paren {\frac {-x^{m - 1} \cos a x} a + \frac {m - 1} a \int x^{m - 2} \cos a x \rd x}\) from $(1)$
\(\ds \) \(=\) \(\ds \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x\) simplifying

$\blacksquare$


Also see


Sources

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