Primitive of Cosine of a x over Power of x

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Theorem

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


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \cos a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -a \sin a x\) Derivative of $\cos a x$


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {x^n}\)
\(\ds \) \(=\) \(\ds x^{-n}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {x^{-n + 1} } {- n + 1}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {\paren {n - 1} x^{n - 1} }\) simplifying


Then:

\(\ds \int \frac {\cos a x} {x^n} \rd x\) \(=\) \(\ds \cos a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {-a \sin a x} \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {-\cos a x} {\paren {n - 1} x^{n - 1} } - \frac a {n - 1} \int \frac {\sin a x} {x^{n - 1} } \rd x\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


Sources

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