Primitive of Sine of a x over Power of x

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

With a view to expressing the primitive in the form:

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

let:

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


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:

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

$\blacksquare$


Also see


Sources