Odd positive power of sine function - Revision history

Vipul: /* Definition */

2011-09-04T14:41:05Z

Definition

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	where <math>n</math> is an odd positive integer, i.e., <math>n = 2k + 1</math> for <math>k</math> a nonnegative integer.		where <math>n</math> is an odd positive integer, i.e., <math>n = 2k + 1</math> for <math>k</math> a nonnegative integer.

			In other words, the function is the [[defining ingredient::composite of two functions\|composite]] of an [[defining ingredient::odd positive power function]] and the [[defining ingredient::sine function]].

	==Integration==		==Integration==

Vipul: moved Odd positive powers of sine function to Odd positive power of sine function

2011-09-04T14:40:03Z

moved Odd positive powers of sine function to Odd positive power of sine function

← Older revision	Revision as of 14:40, 4 September 2011
(No difference)

Vipul: /* First antiderivative: as a polynomial in cosine */

2011-09-04T14:34:30Z

First antiderivative: as a polynomial in cosine

← Older revision		Revision as of 14:34, 4 September 2011
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	\|-		\|-
	\| 2 \|\| 5 \|\| [[fifth power of sine function]] \|\| <math> - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C</math>		\| 2 \|\| 5 \|\| [[fifth power of sine function]] \|\| <math> - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C</math>
			\|-
			\| 3 \|\| 7 \|\| [[seventh power of sine function]] \|\| <math>\frac{\cos^7x}{7} - \frac{3 \cos^5x}{5} + \cos^3x - \cos x + C</math>
	\|}		\|}

Vipul: /* First antiderivative: as a polynomial in cosine */

2011-09-04T14:32:35Z

First antiderivative: as a polynomial in cosine

← Older revision		Revision as of 14:32, 4 September 2011
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	</toggledisplay>		</toggledisplay>

			Note that in all instances, the answer is an [[odd polynomial]] of the cosine function.

	We consider the integration in some small cases:		We consider the integration in some small cases:
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	! <math>k</math> !! <math>n = 2k + 1</math> !! <math>\sin^n</math> !! Antiderivative as a polynomial in <math>\cos</math>		! <math>k</math> !! <math>n = 2k + 1</math> !! <math>\sin^n</math> !! Antiderivative as a polynomial in <math>\cos</math>
	\|-		\|-
	\| 0 \|\| 1 \|\| [[sine function]] \|\| <math>-\cos x + C</math>		\| 0 \|\| 1 \|\| [[sine function]] \|\| <math>\! -\cos x + C</math>
	\|-		\|-
	\| 1 \|\| 3 \|\| [[sine-cubed function]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>		\| 1 \|\| 3 \|\| [[sine-cubed function]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>

Vipul: /* First antiderivative: as a polynomial in cosine */

2011-09-04T14:31:38Z

First antiderivative: as a polynomial in cosine

← Older revision		Revision as of 14:31, 4 September 2011
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	\| 1 \|\| 3 \|\| [[sine-cubed function]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>		\| 1 \|\| 3 \|\| [[sine-cubed function]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>
	\|-		\|-
	\| 2 \|\| 5 \|\| [[fifth power of sine function]] \|\| - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C</math>		\| 2 \|\| 5 \|\| [[fifth power of sine function]] \|\| <math> - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C</math>
	\|}		\|}

Vipul: Created page with "==Definition== This page is about functions of the form: $x \mapsto (\sin x)^n$ where $n$ is an odd positive integer, i.e., $n = 2k + 1$ for <..."

2011-09-04T14:30:13Z

Created page with "==Definition== This page is about functions of the form: <math>x \mapsto (\sin x)^n</math> where <math>n</math> is an odd positive integer, i.e., <math>n = 2k + 1</math> for <..."

New page

==Definition==

This page is about functions of the form:

<math>x \mapsto (\sin x)^n</math>

where <math>n</math> is an odd positive integer, i.e., <math>n = 2k + 1</math> for <math>k</math> a nonnegative integer.

==Integration==

===First antiderivative: as a polynomial in cosine===

We consider <math>n = 2k + 1</math>, <math>k</math> a nonnegative integer:

<math>\int \sin^{2k + 1} x \, dx</math>

Rewrite <math>\sin^{2k + 1}x = \sin^{2k}x \sin x = (1 - \cos^2x)^k \sin x</math>. We get:

<math>\int (1 - \cos^2x)^k(\sin x) \, dx</math>

Set <math>u = \cos x</math>, and we get:

<math>\int -(1 - u^2)^k \, du</math>

This is a polynomial integration in <math>u</math>. After obtaining the answer, we plug back <math>u = \cos x</math>.

Here is the general integration in terms of binomial coefficients: <toggledisplay>

<math>\int -\sum_{r=0}^k (-1)^r \binom{k}{r} u^{2r} \, du</math>

Integrating term-wise, we get:

<math>\sum_{r=0}^k \left[\frac{(-1)^{r + 1}}{2r + 1} \binom{k}{r} u^{2r + 1}\right] + C</math>

Plugging back <math>u = \cos x</math>, we get:

<math>\sum_{r=0}^k \left[\frac{(-1)^{r + 1}}{2r + 1} \binom{k}{r} \cos^{2r + 1}x\right] + C</math>

</toggledisplay>

We consider the integration in some small cases:

{| class="sortable" border="1"
! <math>k</math> !! <math>n = 2k + 1</math> !! <math>\sin^n</math> !! Antiderivative as a polynomial in <math>\cos</math>
|-
| 0 || 1 || [[sine function]] || <math>-\cos x + C</math>
|-
| 1 || 3 || [[sine-cubed function]] || <math>\frac{\cos^3x}{3} - \cos x + C</math>
|-
| 2 || 5 || [[fifth power of sine function]] || - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C</math>
|}

Odd positive power of sine function - Revision history

Vipul: /* Definition */

Vipul: moved Odd positive powers of sine function to Odd positive power of sine function

Vipul: /* First antiderivative: as a polynomial in cosine */

Vipul: /* First antiderivative: as a polynomial in cosine */

Vipul: /* First antiderivative: as a polynomial in cosine */

Vipul: Created page with "==Definition== This page is about functions of the form: x \mapsto (\sin x)^n where n is an odd positive integer, i.e., n = 2k + 1 for <..."

Vipul: Created page with "==Definition== This page is about functions of the form: $x \mapsto (\sin x)^n$ where $n$ is an odd positive integer, i.e., $n = 2k + 1$ for <..."