Reduction Formula for Integral of Power of Sine/Corollary
Corollary to Reduction Formula for Integral of Power of Sine
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let $a \in \R_{\ne 0}$ be a non-zero real number
Then:
- $\ds \int \sin^n a x \rd x = \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x - \dfrac {\sin^{n - 1} a x \cos a x} {a n}$
is a reduction formula for $\ds \int \sin^n a x \rd x$.
Proof
| \(\ds \int \sin^n a x \rd x\) | \(=\) | \(\ds \frac 1 a \int \sin^n a x \map \rd {a x}\) | Primitive of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \paren {\dfrac {n - 1} n \int \sin^{n - 2} a x \map \rd {a x} - \dfrac {\sin^{n - 1} a x \cos a x} n}\) | Reduction Formula for Integral of Power of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \paren {a \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x} - \dfrac {\sin^{n - 1} a x \cos a x} {a n}\) | Primitive of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x - \dfrac {\sin^{n - 1} a x \cos a x} {a n}\) | simplification |
$\blacksquare$
Also see
- Primitive of $\cos^n a x$
- Primitive of $\tan^n a x$
- Primitive of $\cot^n a x$
- Primitive of $\sec^n a x$
- Primitive of $\csc^n a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.366$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $60$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae