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Lösung 2.2:3a

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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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
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{{Displayed math||<math>\int \left( \begin{matrix}
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{{Abgesetzte Formel||<math>\int \left( \begin{matrix}
\text{an expression}\\
\text{an expression}\\
\text{in u}
\text{in u}
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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
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{{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
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{{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,
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{{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}

Version vom 13:01, 10. Mär. 2009

The secret behind a successful substitution is to be able to recognize the integral as an expression of the type

an expressionin uudx 

where u=u(x) is the actual substitution. In the integral

2xsinx2dx 

we see that the expression x2 is the argument for the sine function, as the same time as its derivative x2=2x  stands as a factor in front of sine. Therefore, if we set u=x2, the integral, the integral will be of the form

usinudx. 

Thus, we can use u=x2 for the substitution,

2xsinx2dx=udu=x2=2xdx=sinudu=−cosu+C=−cosx2+C. 
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:3a“