Reduction Formula for Integral of Power of Sine
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let:
- $I_n := \ds \int \sin^n x \rd x$
Then:
- $I_n = \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n$
is a reduction formula for $\ds \int \sin^n x \rd x$.
Corollary
- $\ds \int \sin^n a x \rd x = \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x - \dfrac {\sin^{n - 1} a x \cos a x} {a n}$
Proof 1
Let $n \ge 2$.
Let:
- $\ds I_n := \int \sin^n x \rd x$
Then:
\(\ds I_n\) | \(=\) | \(\ds \int \sin^n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \sin^{n - 1} x \sin x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \sin^{n - 1} x \map \rd {-\cos x}\) | Derivative of Cosine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x - \int \paren {-\cos x} \map \rd {\sin^{n - 1} x}\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x - \int \paren {-\cos x} \paren {n - 1} \sin^{n - 2} x \cos x \rd x\) | Derivative of Power and Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x + \paren {n - 1} \int \sin^{n - 2} x \cos^2 x \rd x\) | Linear Combination of Primitives and rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x + \paren {n - 1} \int \sin^{n - 2} x \paren {1 - \sin^2 x} \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x + \paren {n - 1} \int \sin^{n - 2} x \rd x - \paren {n - 1} \int \sin^n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds - \sin^{n - 1} x \cos x + \paren {n - 1} I_{n - 2} - \paren {n - 1} I_n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds n I_n\) | \(=\) | \(\ds - \sin^{n - 1} x \cos x + \paren {n - 1} I_{n - 2}\) | adding $\paren {n - 1} I_n$ to both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds I_n\) | \(=\) | \(\ds \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n\) | dividing by $n$ and rearranging |
thus demonstrating the identity for all $n \ge 2$.
When $n = 1$ this degenerates to:
\(\ds \int \sin x \rd x\) | \(=\) | \(\ds \dfrac 0 1 \int \frac 1 {\sin x} \rd x - \dfrac {\sin^0 x \cos x} 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 - 1 \cdot \cos x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\cos x\) |
From Primitive of Sine Function this shows that the identity still holds for $n = 1$.
$\blacksquare$
Proof 2
With a view to expressing the problem 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:
\(\ds u\) | \(=\) | \(\ds \sin^{n - 1} x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \paren {n - 1} \sin ^{n - 2} x \cos x\) | Chain Rule for Derivatives, Derivative of Sine Function, Derivative of Power |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sin x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds -\cos x\) | Primitive of Sine Function |
Then:
\(\ds \int \sin^n x \rd x\) | \(=\) | \(\ds \int \sin^{n - 1} x \sin x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin^{n - 1} x \paren {-\cos x} - \int \paren {-\cos x} \paren {\paren {n - 1} \sin^{n - 2} x \cos x} \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {n - 1} \sin^{n - 2} x \cos^2 x \rd x - \sin^{n - 1} x \cos x\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {n - 1} \sin^{n - 2} x \paren {1 - \sin^2 x} \rd x - \sin^{n - 1} x \cos x\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {n - 1} \int \sin^{n - 2} x \rd x - \paren {n - 1} \int \sin^n x \rd x - \sin^{n - 1} x \cos x\) | Linear Combination of Primitives | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \int \sin^n x \rd x\) | \(=\) | \(\ds \paren {n - 1} \int \sin^{n - 2} x \rd x - \sin^{n - 1} x \cos x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sin^n x \rd x\) | \(=\) | \(\ds \dfrac {n - 1} n \int \sin^{n - 2} x \rd x - \dfrac {\sin^{n - 1} x \cos x} n\) | dividing both sides by $n$ |
$\blacksquare$
Also see
- Reduction Formula for Integral of Power of Cosine
- Reduction Formula for Integral of Power of Tangent
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.12$: Wallis's Product
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals: Reduction Formulae