Integrals - Revision history

Brett at 00:21, 25 May 2026

2026-05-25T00:21:31Z

← Older revision		Revision as of 17:21, 24 May 2026
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			== Single Integrals ==

			=== Space Curves: ===
			{{indent\|width=1em}}'''Arclength:'''  <math>\int\limits_{a}^{b}\left \Vert \overrightarrow{r}(t) \right \\| dt</math></div>
			{{indent\|width=1em}}'''Scalar Line Integral:'''  <math>\int_Cf ds = \int\limits_{a}^{b} f(\overrightarrow{r}(t))*\left \Vert \overrightarrow{r}(t)' \right \\| dt</math></div>
			<br>
			== Double Integrals ==

			== Triple Integrals ==

			== Integral Theorems ==

	== Integral Forms & Antiderivatives ==		== Integral Forms & Antiderivatives ==

Brett: /* Forms Involving: \sqrt{2au - u^2},\ a > 02au−u2, a>0 */

2026-05-24T23:41:09Z

Forms Involving: \sqrt{2au - u^2},\ a > 02au−u2, a>0

Brett at 23:37, 24 May 2026

2026-05-24T23:37:09Z

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 # <math>\int \sin^n u\cos^m u\, du = -\frac{\sin^{n-1}u\cos^{m+1}u}{n+m} + \frac{n-1}{n+m}\int \sin^{n-2}u\cos^m u\, du</math>
 {{indent|width=11em}}&nbsp;<math>= \frac{\sin^{n+1}u\cos^{m-1}u}{n+m} + \frac{m-1}{n+m}\int \sin^n u\cos^{m-2}u\, du</math></div>
 == To Do ==
 *Antiderivative DB search system.
 *Find a use for the white-space on the right of the page.
 *Integral Isolating System for easy reference/copying.

Brett: working indent system

2026-05-24T23:32:26Z

working indent system

← Older revision		Revision as of 16:32, 24 May 2026
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	# <math>\int u^n\cos u\, du = u^n\sin u - n\int u^{n-1}\sin u\, du</math>		# <math>\int u^n\cos u\, du = u^n\sin u - n\int u^{n-1}\sin u\, du</math>
	# <math>\int \sin^n u\cos^m u\, du = -\frac{\sin^{n-1}u\cos^{m+1}u}{n+m} + \frac{n-1}{n+m}\int \sin^{n-2}u\cos^m u\, du</math>		# <math>\int \sin^n u\cos^m u\, du = -\frac{\sin^{n-1}u\cos^{m+1}u}{n+m} + \frac{n-1}{n+m}\int \sin^{n-2}u\cos^m u\, du</math>
	{{indent\|40}}		{{indent\|width=11em}} <math>= \frac{\sin^{n+1}u\cos^{m-1}u}{n+m} + \frac{m-1}{n+m}\int \sin^n u\cos^{m-2}u\, du</math></div>
	<math>= \frac{\sin^{n+1}u\cos^{m-1}u}{n+m} + \frac{m-1}{n+m}\int \sin^n u\cos^{m-2}u\, du</math>
	</div>
	== To Do ==		== To Do ==
	*Antiderivative DB search system.		*Antiderivative DB search system.
	*Find a use for the white-space on the right of the page.		*Find a use for the white-space on the right of the page.
	*Integral Isolating System for easy reference/copying.		*Integral Isolating System for easy reference/copying.

Brett at 23:24, 24 May 2026

2026-05-24T23:24:16Z

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 # <math>\int \frac{du}{u^n\sqrt{a + bu}} = -\frac{\sqrt{a + bu}}{a(n-1)u^{n-1}} - \frac{b(2n-3)}{2a(n-1)}\int \frac{du}{u^{n-1}\sqrt{a + bu}}</math>
 <br>
 == To Do ==
 *Antiderivative DB search system.
 *Find a use for the white-space on the right of the page.
 *Integral Isolating System for easy reference/copying.

Brett at 22:45, 24 May 2026

2026-05-24T22:45:01Z

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 == Integral Forms & Antiderivatives ==
 === Basic Forms ===
 # <math>\int u \, dv = uv - \int v \, du</math><br>
 # <math>\int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad (n \neq -1)</math><br>
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 # <math>\int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln \left| \frac{u-a}{u+a} \right| + C</math><br>
 <br>
 == To Do ==
 *Antiderivative DB search system.
 *Find a use for the white-space on the right of the page.

Brett: /* To Do */

2026-05-24T22:25:04Z

To Do

← Older revision		Revision as of 15:25, 24 May 2026
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	== To Do ==		== To Do ==
	*Antiderivative DB search system.		*Antiderivative DB search system.
	*Find a use for the white-space on the ~~left~~ of the page.		*Find a use for the white-space on the right of the page.

Brett: /* To Do */

2026-05-24T22:24:45Z

To Do

← Older revision		Revision as of 15:24, 24 May 2026
Line 27:		Line 27:
	== To Do ==		== To Do ==
	*Antiderivative DB search system.		*Antiderivative DB search system.
			*Find a use for the white-space on the left of the page.

Brett at 22:23, 24 May 2026

2026-05-24T22:23:21Z

← Older revision		Revision as of 15:23, 24 May 2026
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	== Integral Forms & Antiderivatives ==		== Integral Forms & Antiderivatives ==
	=== Basic Forms ===		=== Basic Forms ===

	# <math>\int u \, dv = uv - \int v \, du</math><br>		# <math>\int u \, dv = uv - \int v \, du</math><br>
	# <math>\int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad (n \neq -1)</math><br>		# <math>\int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad (n \neq -1)</math><br>
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	# <math>\int \frac{du}{a^2 - u^2} = \frac{1}{2a} \ln \left\| \frac{u+a}{u-a} \right\| + C</math><br>		# <math>\int \frac{du}{a^2 - u^2} = \frac{1}{2a} \ln \left\| \frac{u+a}{u-a} \right\| + C</math><br>
	# <math>\int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln \left\| \frac{u-a}{u+a} \right\| + C</math><br>		# <math>\int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln \left\| \frac{u-a}{u+a} \right\| + C</math><br>
			<br>
			=== Forms Involving <math>\sqrt{a}</math> ===

			== To Do ==
			*Antiderivative DB search system.

Brett: basic integral forms

2026-05-24T22:19:05Z

basic integral forms

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-<math>E=mc^2</math>
+== Integral Forms & Antiderivatives ==
-<big>
+=== Basic Forms ===
-<big><math display="block">\int\frac{e^3/x}{x^2}\, dx</math></big></big>

← Older revision		Revision as of 16:41, 24 May 2026
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	# <math>\int \frac{u\, du}{\sqrt{a + bu}} = \frac{2}{3b^2}(bu - 2a)\sqrt{a + bu} + C</math>		# <math>\int \frac{u\, du}{\sqrt{a + bu}} = \frac{2}{3b^2}(bu - 2a)\sqrt{a + bu} + C</math>
	# <math>\int \frac{u^2\, du}{\sqrt{a + bu}} = \frac{2}{15b^3}(8a^2 + 3b^2u^2 - 4abu)\sqrt{a + bu} + C</math>		# <math>\int \frac{u^2\, du}{\sqrt{a + bu}} = \frac{2}{15b^3}(8a^2 + 3b^2u^2 - 4abu)\sqrt{a + bu} + C</math>
	# <math>\int \frac{du}{u\sqrt{a + bu}} = \begin{cases} \dfrac{1}{\sqrt{a}}\ln\left\|\dfrac{\sqrt{a + bu} - \sqrt{a}}{\sqrt{a + bu} + \sqrt{a}}\right\| + C & \text{if } a > 0 \\[6pt] \dfrac{2}{\sqrt{-a}}\tan^{-1}\sqrt{\dfrac{a + bu}{-a}} + C & \text{if } a < 0 \end{cases}</math>		# <math>\int \frac{du}{u\sqrt{a + bu}} = \begin{cases} \dfrac{1}{\sqrt{a}}\ln\left\|\dfrac{\sqrt{a + bu} - \sqrt{a}}{\sqrt{a + bu} + \sqrt{a}}\right\| + C & (\text{if } a > 0) \\[6pt] \dfrac{2}{\sqrt{-a}}\tan^{-1}\sqrt{\dfrac{a + bu}{-a}} + C & (\text{if } a < 0) \end{cases}</math>
	# <math>\int \frac{\sqrt{a + bu}}{u}\, du = 2\sqrt{a + bu} + a\int \frac{du}{u\sqrt{a + bu}}</math>		# <math>\int \frac{\sqrt{a + bu}}{u}\, du = 2\sqrt{a + bu} + a\int \frac{du}{u\sqrt{a + bu}}</math>
	# <math>\int \frac{\sqrt{a + bu}}{u^2}\, du = -\frac{\sqrt{a + bu}}{u} + \frac{b}{2}\int \frac{du}{u\sqrt{a + bu}}</math>		# <math>\int \frac{\sqrt{a + bu}}{u^2}\, du = -\frac{\sqrt{a + bu}}{u} + \frac{b}{2}\int \frac{du}{u\sqrt{a + bu}}</math>
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	# <math>\int u^n\sin u\, du = -u^n\cos u + n\int u^{n-1}\cos u\, du</math>		# <math>\int u^n\sin u\, du = -u^n\cos u + n\int u^{n-1}\cos u\, du</math>
	# <math>\int u^n\cos u\, du = u^n\sin u - n\int u^{n-1}\sin u\, du</math>		# <math>\int u^n\cos u\, du = u^n\sin u - n\int u^{n-1}\sin u\, du</math>
	# <math>\int \sin^n u\cos^m u\, du = -\frac{\sin^{n-1}u\cos^{m+1}u}{n+m} + \frac{n-1}{n+m}\int \sin^{n-2}u\cos^m u\, du</math>		# <math>\int \sin^n u\cos^m u\, du = -\frac{\sin^{n-1}u\cos^{m+1}u}{n+m} + \frac{n-1}{n+m}\int \sin^{n-2}u\cos^m u\, du</math>
	{{indent\|width=11em}} <math>= \frac{\sin^{n+1}u\cos^{m-1}u}{n+m} + \frac{m-1}{n+m}\int \sin^n u\cos^{m-2}u\, du</math></div>		{{indent\|width=11em}} <math>= \frac{\sin^{n+1}u\cos^{m-1}u}{n+m} + \frac{m-1}{n+m}\int \sin^n u\cos^{m-2}u\, du</math></div>
	<br>		<br>
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	# <math>\int u\cos^{-1}u\, du = \frac{2u^2 - 1}{4}\cos^{-1}u - \frac{u\sqrt{1 - u^2}}{4} + C</math>		# <math>\int u\cos^{-1}u\, du = \frac{2u^2 - 1}{4}\cos^{-1}u - \frac{u\sqrt{1 - u^2}}{4} + C</math>
	# <math>\int u\tan^{-1}u\, du = \frac{u^2 + 1}{2}\tan^{-1}u - \frac{u}{2} + C</math>		# <math>\int u\tan^{-1}u\, du = \frac{u^2 + 1}{2}\tan^{-1}u - \frac{u}{2} + C</math>
	# <math>\int u^n\sin^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\sin^{-1}u - \int \frac{u^{n+1}\, du}{\sqrt{1 - u^2}}\right], \quad n \neq -1</math>		# <math>\int u^n\sin^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\sin^{-1}u - \int \frac{u^{n+1}\, du}{\sqrt{1 - u^2}}\right], \quad (n \neq -1)</math>
	# <math>\int u^n\cos^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\cos^{-1}u + \int \frac{u^{n+1}\, du}{\sqrt{1 - u^2}}\right], \quad n \neq -1</math>		# <math>\int u^n\cos^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\cos^{-1}u + \int \frac{u^{n+1}\, du}{\sqrt{1 - u^2}}\right], \quad (n \neq -1)</math>
	# <math>\int u^n\tan^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\tan^{-1}u - \int \frac{u^{n+1}\, du}{1 + u^2}\right], \quad n \neq -1</math>		# <math>\int u^n\tan^{-1}u\, du = \frac{1}{n+1}\left[u^{n+1}\tan^{-1}u - \int \frac{u^{n+1}\, du}{1 + u^2}\right], \quad (n \neq -1)</math>
	</br>		</br>