Lösung 2.2:3a - Versionsgeschichte

Sekretariat1 um 12:36, 27. Aug. 2009

2009-08-27T12:36:05Z

← Nächstältere Version		Version vom 12:36, 27. Aug. 2009
Zeile 1:		Zeile 1:
-	~~Wir sehen, dass die~~ Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ~~ist und daher~~ substituieren wir <math>u=x^2</math>.	+	Die Ableitung von <math>x^2</math> ist <math>\bigl(x^2\bigr)'=2x</math>. Daher substituieren wir <math>u=x^2</math>.

	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx</math>}}		{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx</math>}}

Sekretariat1 um 12:30, 27. Aug. 2009

2009-08-27T12:30:30Z

← Nächstältere Version		Version vom 12:30, 27. Aug. 2009
Zeile 1:		Zeile 1:
-	Wir sehen, dass die Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ist und daher substituieren wir <math>u=x^2</math>,	+	Wir sehen, dass die Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ist und daher substituieren wir <math>u=x^2</math>.

-	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx~~\,\textrm{.}~~</math>}}	+	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx</math>}}

	Wir erhalten		Wir erhalten

Moni um 09:51, 22. Aug. 2009

2009-08-22T09:51:35Z

← Nächstältere Version		Version vom 09:51, 22. Aug. 2009
Zeile 1:		Zeile 1:
-	Wir sehen dass die Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ist, und daher ~~machen~~ wir ~~die Substitution~~ <math>u=x^2</math>,	+	Wir sehen, dass die Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ist und daher substituieren wir <math>u=x^2</math>,

	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}

-	Wir erhalten,	+	Wir erhalten

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}

Martin um 13:17, 5. Mai 2009

2009-05-05T13:17:50Z

← Nächstältere Version		Version vom 13:17, 5. Mai 2009
Zeile 1:		Zeile 1:
-	~~The secret behind a successful substitution is to be able to recognize the integral as an expression of the type~~	+	Wir sehen dass die Ableitung von <math>x^2</math>, <math>\bigl(x^2\bigr)'=2x</math> ist, und daher machen wir die Substitution <math>u=x^2</math>,
-		+
-	~~{{Abgesetzte Formel\|\|~~<math>~~\int \left( \begin{matrix}~~	+
-	~~\text{an expression}\\~~	+
-	~~\text{in u}~~	+
-	~~\end{matrix}\right)\cdot u'\,dx\,,</math>}}~~	+
-		+
-	~~where <math>u=u(x)</math> is the actual substitution. In the integral~~	+
-		+
-	~~{{Abgesetzte Formel\|\|<math>\int 2x\sin~~ x^2~~\,dx~~</math>}}	+
-		+
-	~~we see that the expression <math>x^2</math> is the argument for the sine function~~, ~~as the same time as its derivative~~ <math>\bigl(x^2\bigr)'=2x</math> ~~stands as a factor in front of sine. Therefore~~, ~~if we set~~ <math>u=x^2</math>, ~~the integral, the integral will be of the form~~	+

	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}

-	~~Thus, we can use <math>u=x^2</math> for the substitution~~,	+	Wir erhalten,

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}

Tekbot: Solution 2.2:3a moved to Lösung 2.2:3a: Robot: moved page

2009-03-11T10:22:35Z

Solution 2.2:3a moved to Lösung 2.2:3a: Robot: moved page

← Nächstältere Version

Version vom 10:22, 11. Mär. 2009

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

2009-03-10T13:01:44Z

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

← Nächstältere Version		Version vom 13:01, 10. Mär. 2009
Zeile 1:		Zeile 1:
	The secret behind a successful substitution is to be able to recognize the integral as an expression of the type		The secret behind a successful substitution is to be able to recognize the integral as an expression of the type

-	{{~~Displayed math~~\|\|<math>\int \left( \begin{matrix}	+	{{Abgesetzte Formel\|\|<math>\int \left( \begin{matrix}
	\text{an expression}\\		\text{an expression}\\
	\text{in u}		\text{in u}
Zeile 8:		Zeile 8:
	where <math>u=u(x)</math> is the actual substitution. In the integral		where <math>u=u(x)</math> is the actual substitution. In the integral

-	{{~~Displayed math~~\|\|<math>\int 2x\sin x^2\,dx</math>}}	+	{{Abgesetzte Formel\|\|<math>\int 2x\sin x^2\,dx</math>}}

	we see that the expression <math>x^2</math> is the argument for the sine function, as the same time as its derivative <math>\bigl(x^2\bigr)'=2x</math> stands as a factor in front of sine. Therefore, if we set <math>u=x^2</math>, the integral, the integral will be of the form		we see that the expression <math>x^2</math> is the argument for the sine function, as the same time as its derivative <math>\bigl(x^2\bigr)'=2x</math> stands as a factor in front of sine. Therefore, if we set <math>u=x^2</math>, the integral, the integral will be of the form

-	{{~~Displayed math~~\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}

	Thus, we can use <math>u=x^2</math> for the substitution,		Thus, we can use <math>u=x^2</math> for the substitution,

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	\int 2x\sin x^2\,dx		\int 2x\sin x^2\,dx
	&=\left\{\begin{align}		&=\left\{\begin{align}

Tek um 14:04, 28. Okt. 2008

2008-10-28T14:04:33Z

← Nächstältere Version		Version vom 14:04, 28. Okt. 2008
Zeile 1:		Zeile 1:
	The secret behind a successful substitution is to be able to recognize the integral as an expression of the type		The secret behind a successful substitution is to be able to recognize the integral as an expression of the type

		+	{{Displayed math\|\|<math>\int \left( \begin{matrix}
		+	\text{an expression}\\
		+	\text{in u}
		+	\end{matrix}\right)\cdot u'\,dx\,,</math>}}

-	<math>~~\int{\left( \begin{matrix}~~	+	where <math>u=u(x)</math> is the actual substitution. In the integral
-	~~\text{an}\quad \text{expression} \\~~	+
-	~~\text{in}\quad~~ u \\	+
-	~~\end{matrix} \right)}\centerdot {~~u~~}'\,dx~~</math>,	+

-	~~where~~	+	{{Displayed math\|\|<math>\int 2x\sin x^2\,dx</math>}}
-	<math>~~u=u~~\~~left(~~ x \~~right)~~</math>	+
-	~~is the actual substitution. In the integral~~	+

		+	we see that the expression <math>x^2</math> is the argument for the sine function, as the same time as its derivative <math>\bigl(x^2\bigr)'=2x</math> stands as a factor in front of sine. Therefore, if we set <math>u=x^2</math>, the integral, the integral will be of the form

-	<math>\int~~{2x~~\sin ~~x^{2}~~\,dx}</math>	+	{{Displayed math\|\|<math>\int u'\sin u\,dx\,\textrm{.}</math>}}

		+	Thus, we can use <math>u=x^2</math> for the substitution,

-	~~we see that the expression~~	+	{{Displayed math\|\|<math>\begin{align}
-	~~<math>x^~~{~~2}</math>~~	+	\int 2x\sin x^2\,dx
-	~~is the argument for the sine function, as the same time as its derivative~~	+	&=\left\{\begin{align}
-	~~<math>\left( x^~~{~~2} \right)^{\prime }=2x</~~math>	+	u &= x^2\\[5pt]
-	~~stands as a factor in front of sine. Therefore, if we set~~	+	du &= 2x\,dx
-	~~<math>u=x^{2}</math>, the integral, the integral will be of the form~~	+	\end{align}\right\} = \int{\sin u\,du}\\[5pt]
-		+	&= -\cos u+C = -\cos x^2 + C\,\textrm{.}
-		+	\end{align}</math>}}
-	~~<math>\int{{u}'\sin u\,dx}</math>~~	+
-		+
-		+
-	~~Thus, we can use~~	+
-	~~<math>u=x^{2}</math>~~	+
-	~~for the substitution:~~	+
-		+
-		+
-	<math>\begin{align}	+
-	& \int{2x\sin x^{2}\,dx}=\left\{ \begin{~~matrix~~}	+
-	u=x^{2} \\	+
-	du=2x\,dx \\	+
-	\end{~~matrix~~} \right\}=\int{\sin u\,du} \\	+
-	& =-\cos u+C=-\cos x^{2}+C \\	+
-	\end{align}</math>	+

Ian um 12:51, 20. Okt. 2008

2008-10-20T12:51:36Z

← Nächstältere Version		Version vom 12:51, 20. Okt. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START~~}}	+	The secret behind a successful substitution is to be able to recognize the integral as an expression of the type
-	<~~center~~> ~~[[Image:2_2_3a~~.~~gif]]~~ </~~center~~>	+
-	{{~~NAVCONTENT_STOP~~}}	+
		+	<math>\int{\left( \begin{matrix}
		+	\text{an}\quad \text{expression} \\
		+	\text{in}\quad u \\
		+	\end{matrix} \right)}\centerdot {u}'\,dx</math>,
		+
		+	where
		+	<math>u=u\left( x \right)</math>
		+	is the actual substitution. In the integral
		+
		+
		+	<math>\int{2x\sin x^{2}\,dx}</math>
		+
		+
		+	we see that the expression
		+	<math>x^{2}</math>
		+	is the argument for the sine function, as the same time as its derivative
		+	<math>\left( x^{2} \right)^{\prime }=2x</math>
		+	stands as a factor in front of sine. Therefore, if we set
		+	<math>u=x^{2}</math>, the integral, the integral will be of the form
		+
		+
		+	<math>\int{{u}'\sin u\,dx}</math>
		+
		+
		+	Thus, we can use
		+	<math>u=x^{2}</math>
		+	for the substitution:
		+
		+
		+	<math>\begin{align}
		+	& \int{2x\sin x^{2}\,dx}=\left\{ \begin{matrix}
		+	u=x^{2} \\
		+	du=2x\,dx \\
		+	\end{matrix} \right\}=\int{\sin u\,du} \\
		+	& =-\cos u+C=-\cos x^{2}+C \\
		+	\end{align}</math>

Tekbot: Lösning 2.2:3a moved to Solution 2.2:3a: Robot: moved page

2008-09-16T14:43:01Z

Lösning 2.2:3a moved to Solution 2.2:3a: Robot: moved page

← Nächstältere Version

Version vom 14:43, 16. Sep. 2008

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

2008-08-21T08:01:14Z

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

← Nächstältere Version		Version vom 08:01, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_2_3a.gif]] </center>	+	<center> [[Image:2_2_3a.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}