Primitive of Sine of a x over Power of x
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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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.365$