Lösung 2.2:1b - Versionsgeschichte

Sekretariat1 um 11:54, 27. Aug. 2009

2009-08-27T11:54:39Z

← Nächstältere Version		Version vom 11:54, 27. Aug. 2009
Zeile 20:		Zeile 20:
	{{Abgesetzte Formel\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C</math>}}		{{Abgesetzte Formel\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C</math>}}

-	Wir schreiben nun die Antwort in der Variable <math>x</math>, indem wir die Substitution <math>u=x^{2}+3</math> ausführen	+	Wir schreiben nun die Antwort in der Variable <math>x</math>, indem wir die Substitution <math>u=x^{2}+3</math> ausführen.

	{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,</math>}}		{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,</math>}}

Sekretariat1 um 11:54, 27. Aug. 2009

2009-08-27T11:54:09Z

Martin um 12:44, 5. Mai 2009

2009-05-05T12:44:01Z

← Nächstältere Version		Version vom 12:44, 5. Mai 2009
Zeile 1:		Zeile 1:
-	~~For an indefinite integral~~, we do not need to take account of the limits of integration when substituting variables, but at the end, when the integral has been calculated, we do need to change back to the variable <math>x</math> (because the original integral was expressed in <math>x</math>).	+	Um ein unbestimmtes Integral zu berechnen, besteht kein Bedarf die Grenzen zu ändern. Wir müssen aber nach der Integration wieder zurück zu der Variable <math>x</math> gehen.

-	~~If we start by looking at the integration element <math>du</math>, the relation between~~ <math>dx</math> ~~and~~ <math>du</math> ~~reads~~	+	Das Verhältnis zwischen <math>dx</math> und <math>du</math> lautet

	{{Abgesetzte Formel\|\|<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}		{{Abgesetzte Formel\|\|<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}

-	~~which can be written as~~	+	oder,

	{{Abgesetzte Formel\|\|<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}

-	~~The expression~~ <math>x\,dx</math> ~~is present as a factor in the integral~~, ~~and so everything is there for the substitution~~ <math>u=x^{2}+3</math>,	+	Nachdem <math>x\,dx</math> ein Faktor im Integrand ist, können wir die Substitution <math>u=x^{2}+3</math> direkt ausführen,

	{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}		{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}
Zeile 16:		Zeile 16:
	\end{align}\right\} = \int u^5\cdot\tfrac{1}{2}\,du\,\textrm{.}</math>}}		\end{align}\right\} = \int u^5\cdot\tfrac{1}{2}\,du\,\textrm{.}</math>}}

-	~~The result on the right-hand side is a standard integral~~, ~~which we integrate directly~~,	+	Dies ist nun ein Standardintegral, das wir direkt berechnen,

	{{Abgesetzte Formel\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}

-	~~We write the answer expressed~~ in <math>x</math> ~~by substituting back~~ <math>u=x^{2}+3</math>,	+	Wir schreiben nun die Antwort in der Variable <math>x</math> indem wir die Substitution <math>u=x^{2}+3</math> ausführen,

	{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}		{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}

-	~~where~~ <math>C</math> ~~is an arbitrary constant~~.	+	Wo <math>C</math> eine beliebige Konstante ist.

-		+	Hinweis: Wir können natürlich unsere Rechnungen überprüfen, indem wir <math>\tfrac{1}{12}( x^{2}+3)^6+C</math> ableiten, und sehen ob wir <math>(x^2+3)^5x\,</math> erhalten.
-	~~Note~~: ~~It is possible to check the answer by differentiating~~ <math>\tfrac{1}{12}( x^{2}+3)^6+C</math> ~~and seeing that we get back the integrand~~ <math>(x^2+3)^5x\,</math>.	+

Tekbot: Solution 2.2:1b moved to Lösung 2.2:1b: Robot: moved page

2009-03-11T10:20:35Z

Solution 2.2:1b moved to Lösung 2.2:1b: Robot: moved page

← Nächstältere Version

Version vom 10:20, 11. Mär. 2009

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

2009-03-10T13:00:44Z

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tek um 12:43, 28. Okt. 2008

2008-10-28T12:43:25Z

← Nächstältere Version		Version vom 12:43, 28. Okt. 2008
Zeile 1:		Zeile 1:
-	For an indefinite integral, we do not need to take account of the limits of integration when substituting variables, but at the end, when the integral has been calculated, we do need to change back to the variable	+	For an indefinite integral, we do not need to take account of the limits of integration when substituting variables, but at the end, when the integral has been calculated, we do need to change back to the variable <math>x</math> (because the original integral was expressed in <math>x</math>).
-	<math>x</math>	+
-	(because the original integral was expressed in	+
-	<math>x</math>).	+

-	If we start by looking at the integration element	+	If we start by looking at the integration element <math>du</math>, the relation between <math>dx</math> and <math>du</math> reads
-	<math>du</math>, the relation between	+
-	<math>dx</math>	+
-	and	+
-	<math>du</math>	+
-	reads	+

-		+	{{Displayed math\|\|<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}
-	<math>du={u}'~~\left~~( x ~~\right~~)\,dx=~~\left~~( x^{2}+3 ~~\right~~)~~^{\prime }~~\,dx=2x\,dx</math>,	+

	which can be written as		which can be written as

		+	{{Displayed math\|\|<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}

-	~~<math>x\,dx=\frac{1}{2}\,du</math>~~	+	The expression <math>x\,dx</math> is present as a factor in the integral, and so everything is there for the substitution <math>u=x^{2}+3</math>,
-		+
-		+
-	The expression	+
-	<math>x\,dx</math>	+
-	is present as a factor in the integral, and so everything is there for the substitution	+
-	<math>u=x^{2}+3</math>	+

-		+	{{Displayed math\|\|<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}
-		+	u &= x^2+3\\[5pt]
-	<math>\int~~{\left~~( x^{2}+3 ~~\right~~)}^~~{5}x~~\,dx=\left\{ \begin{~~matrix~~}	+	du &= 2x\,dx
-	u=x^{2}+3 \\	+	\end{align}\right\} = \int u^5\cdot\tfrac{1}{2}\,du\,\textrm{.}</math>}}
-	du=2x\,dx \\	+
-	\end{~~matrix~~} \right\}=\int{u^{5}}\~~centerdot~~ \~~frac~~{1}{2}\,du</math>	+
-		+

	The result on the right-hand side is a standard integral, which we integrate directly,		The result on the right-hand side is a standard integral, which we integrate directly,

		+	{{Displayed math\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}

-	~~<math>\frac{1}{2}\int{u^{5}}\,du=\frac{1}{2}\centerdot \frac{u^{6}}{6}+C</math>~~	+	We write the answer expressed in <math>x</math> by substituting back <math>u=x^{2}+3</math>,
-		+
-		+
-	We write the answer expressed in	+
-	<math>x~~\text{ }~~</math>	+
-	by substituting back	+
-	<math>u=x^{2}+3</math>,	+

		+	{{Displayed math\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}

-	<math>~~\int{\left( x^{2}+3 \right)}^{5}x\,dx=\frac{\left( x^{2}+3 \right)^{6}}{12}+~~C</math>	+	where <math>C</math> is an arbitrary constant.

-	where
-	<math>C</math>
-	is an arbitrary constant.

-	~~NOTE~~: it is possible to check the answer by differentiating	+	Note: It is possible to check the answer by differentiating <math>\tfrac{1}{12}( x^{2}+3)^6+C</math> and seeing that we get back the integrand <math>(x^2+3)^5x\,</math>.
-	<math>\~~frac~~{1}{12}~~\left~~( x^{2}+3 ~~\right~~)^{6}+C</math>	+
-	and seeing that we get back the integrand	+
-	<math>~~\left~~( x^{2}+3 ~~\right~~)^~~{5}x~~</math>.	+

Ian um 09:48, 19. Okt. 2008

2008-10-19T09:48:04Z

← Nächstältere Version		Version vom 09:48, 19. Okt. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START~~}}	+	For an indefinite integral, we do not need to take account of the limits of integration when substituting variables, but at the end, when the integral has been calculated, we do need to change back to the variable
-	<~~center~~> ~~[[Image:2_2_1b-~~1(2)~~.gif]]~~ </~~center~~>	+	<math>x</math>
-	{{~~NAVCONTENT_STOP~~}}	+	(because the original integral was expressed in
-	{{~~NAVCONTENT_START~~}}	+	<math>x</math>).
-	<~~center~~> ~~[[Image:2_2_1b-~~2(2)~~.gif]]~~ </~~center~~>	+
-	{{~~NAVCONTENT_STOP~~}}	+	If we start by looking at the integration element
		+	<math>du</math>, the relation between
		+	<math>dx</math>
		+	and
		+	<math>du</math>
		+	reads
		+
		+
		+	<math>du={u}'\left( x \right)\,dx=\left( x^{2}+3 \right)^{\prime }\,dx=2x\,dx</math>,
		+
		+	which can be written as
		+
		+
		+	<math>x\,dx=\frac{1}{2}\,du</math>
		+
		+
		+	The expression
		+	<math>x\,dx</math>
		+	is present as a factor in the integral, and so everything is there for the substitution
		+	<math>u=x^{2}+3</math>
		+
		+
		+
		+	<math>\int{\left( x^{2}+3 \right)}^{5}x\,dx=\left\{ \begin{matrix}
		+	u=x^{2}+3 \\
		+	du=2x\,dx \\
		+	\end{matrix} \right\}=\int{u^{5}}\centerdot \frac{1}{2}\,du</math>
		+
		+
		+	The result on the right-hand side is a standard integral, which we integrate directly,
		+
		+
		+	<math>\frac{1}{2}\int{u^{5}}\,du=\frac{1}{2}\centerdot \frac{u^{6}}{6}+C</math>
		+
		+
		+	We write the answer expressed in
		+	<math>x\text{ }</math>
		+	by substituting back
		+	<math>u=x^{2}+3</math>,
		+
		+
		+	<math>\int{\left( x^{2}+3 \right)}^{5}x\,dx=\frac{\left( x^{2}+3 \right)^{6}}{12}+C</math>
		+
		+	where
		+	<math>C</math>
		+	is an arbitrary constant.
		+
		+	NOTE: it is possible to check the answer by differentiating
		+	<math>\frac{1}{12}\left( x^{2}+3 \right)^{6}+C</math>
		+	and seeing that we get back the integrand
		+	<math>\left( x^{2}+3 \right)^{5}x</math>.

Tekbot: Lösning 2.2:1b moved to Solution 2.2:1b: Robot: moved page

2008-09-16T14:41:01Z

Lösning 2.2:1b moved to Solution 2.2:1b: Robot: moved page

← Nächstältere Version

Version vom 14:41, 16. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

2008-08-21T08:00:14Z

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 08:00, 21. Aug. 2008
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	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_2_1b-1(2).gif]] </center>	+	<center> [[Image:2_2_1b-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_2_1b-2(2).gif]] </center>	+	<center> [[Image:2_2_1b-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

Oskar um 09:21, 7. Apr. 2008

2008-04-07T09:21:11Z

← Nächstältere Version		Version vom 09:21, 7. Apr. 2008
Zeile 3:		Zeile 3:
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[Bild:2_2_1b-1(2).gif]] </center>	+	<center> [[Bild:2_2_1b-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

← Nächstältere Version		Version vom 13:00, 10. Mär. 2009
Zeile 3:		Zeile 3:
	If we start by looking at the integration element <math>du</math>, the relation between <math>dx</math> and <math>du</math> reads		If we start by looking at the integration element <math>du</math>, the relation between <math>dx</math> and <math>du</math> reads

-	{{~~Displayed math~~\|\|<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}

	which can be written as		which can be written as

-	{{~~Displayed math~~\|\|<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}

	The expression <math>x\,dx</math> is present as a factor in the integral, and so everything is there for the substitution <math>u=x^{2}+3</math>,		The expression <math>x\,dx</math> is present as a factor in the integral, and so everything is there for the substitution <math>u=x^{2}+3</math>,

-	{{~~Displayed math~~\|\|<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}	+	{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}
	u &= x^2+3\\[5pt]		u &= x^2+3\\[5pt]
	du &= 2x\,dx		du &= 2x\,dx
Zeile 18:		Zeile 18:
	The result on the right-hand side is a standard integral, which we integrate directly,		The result on the right-hand side is a standard integral, which we integrate directly,

-	{{~~Displayed math~~\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}

	We write the answer expressed in <math>x</math> by substituting back <math>u=x^{2}+3</math>,		We write the answer expressed in <math>x</math> by substituting back <math>u=x^{2}+3</math>,

-	{{~~Displayed math~~\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}

	where <math>C</math> is an arbitrary constant.		where <math>C</math> is an arbitrary constant.