Primitive of Power of x by Cosine of a x/Corollary

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Theorem

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

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


Proof

\(\text {(1)}: \quad\) \(\ds \int x^m \cos a x \rd x\) \(=\) \(\ds \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x\) Primitive of Power of x by Cosine of a x
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \int x^{m - 2} \cos a x \rd x\) \(=\) \(\ds \frac {x^{m - 2} \sin a x} a + \frac {\paren {m - 2} x^{m - 3} \cos a x} {a^2} - \frac {\paren {m - 2} \paren {m - 3} } {a^2} \int x^{m - 4} \cos a x \rd x\) setting $m$ equal to $m - 2$
\(\ds \leadsto \ \ \) \(\ds \int x^m \cos a x \rd x\) \(=\) \(\ds \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \paren {\frac {x^{m-2} \sin a x} a + \frac {\paren {m-2} x^{m - 3} \cos a x} {a^2} - \frac {\paren {m - 2} \paren {m - 3} } {a^2} \int x^{m - 4} \cos a x \rd x}\) substituting $(2)$ into $(1)$ above
\(\ds \) \(=\) \(\ds \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {\paren m \paren {m - 1} x^{m - 2} \sin a x} {a^3} - \frac {\paren m \paren {m - 1} \paren {m - 2} x^{m - 3} \cos a x} {a^4} + \frac {\paren m \paren {m - 1} \paren {m - 2} \paren {m - 3} } {a^4} \int x^{m - 4} \cos a x \rd x\) Real Multiplication Distributes over Addition
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^{m + 1} \paren {m^{\underline {k - 1} } \frac {x^{m + 1 - k} } {a^k} \map \sin {x + \frac {\pi} 2 \paren { k - 1 } } }\) Definition of Falling Factorial

$\blacksquare$