Lösung 2.2:1b

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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
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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>).
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<math>x</math>
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(because the original integral was expressed in
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<math>x</math>).
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If we start by looking at the integration element
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If we start by looking at the integration element <math>du</math>, the relation between <math>dx</math> and <math>du</math> reads
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<math>du</math>, the relation between
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<math>dx</math>
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and
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<math>du</math>
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reads
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{{Displayed math||<math>du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,</math>}}
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<math>du={u}'\left( x \right)\,dx=\left( x^{2}+3 \right)^{\prime }\,dx=2x\,dx</math>,
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which can be written as
which can be written as
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{{Displayed math||<math>x\,dx = \tfrac{1}{2}\,du\,\textrm{.}</math>}}
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<math>x\,dx=\frac{1}{2}\,du</math>
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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>,
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The expression
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<math>x\,dx</math>
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is present as a factor in the integral, and so everything is there for the substitution
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<math>u=x^{2}+3</math>
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{{Displayed math||<math>\int (x^2+3)^5x\,dx = \left\{\begin{align}
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u &= x^2+3\\[5pt]
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<math>\int{\left( x^{2}+3 \right)}^{5}x\,dx=\left\{ \begin{matrix}
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du &= 2x\,dx
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u=x^{2}+3 \\
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\end{align}\right\} = \int u^5\cdot\tfrac{1}{2}\,du\,\textrm{.}</math>}}
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du=2x\,dx \\
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\end{matrix} \right\}=\int{u^{5}}\centerdot \frac{1}{2}\,du</math>
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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,
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{{Displayed math||<math>\frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}</math>}}
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<math>\frac{1}{2}\int{u^{5}}\,du=\frac{1}{2}\centerdot \frac{u^{6}}{6}+C</math>
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We write the answer expressed in <math>x</math> by substituting back <math>u=x^{2}+3</math>,
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We write the answer expressed in
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<math>x\text{ }</math>
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by substituting back
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<math>u=x^{2}+3</math>,
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{{Displayed math||<math>\int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,</math>}}
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<math>\int{\left( x^{2}+3 \right)}^{5}x\,dx=\frac{\left( x^{2}+3 \right)^{6}}{12}+C</math>
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where <math>C</math> is an arbitrary constant.
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where
 
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<math>C</math>
 
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is an arbitrary constant.
 
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NOTE: it is possible to check the answer by differentiating
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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>.
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<math>\frac{1}{12}\left( x^{2}+3 \right)^{6}+C</math>
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and seeing that we get back the integrand
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<math>\left( x^{2}+3 \right)^{5}x</math>.
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Version vom 12:43, 28. Okt. 2008

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 \displaystyle x (because the original integral was expressed in \displaystyle x).

If we start by looking at the integration element \displaystyle du, the relation between \displaystyle dx and \displaystyle du reads

\displaystyle du = u'(x)\,dx = (x^2+3)'\,dx = 2x\,dx\,,

which can be written as

\displaystyle x\,dx = \tfrac{1}{2}\,du\,\textrm{.}

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

\displaystyle \int (x^2+3)^5x\,dx = \left\{\begin{align}

u &= x^2+3\\[5pt] du &= 2x\,dx \end{align}\right\} = \int u^5\cdot\tfrac{1}{2}\,du\,\textrm{.}

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

\displaystyle \frac{1}{2}\int u^5\,du = \frac{1}{2}\cdot\frac{u^6}{6} + C\,\textrm{.}

We write the answer expressed in \displaystyle x by substituting back \displaystyle u=x^{2}+3,

\displaystyle \int (x^2+3)^5x\,dx = \frac{(x^2+3)^6}{12}+C\,,

where \displaystyle C is an arbitrary constant.


Note: It is possible to check the answer by differentiating \displaystyle \tfrac{1}{12}( x^{2}+3)^6+C and seeing that we get back the integrand \displaystyle (x^2+3)^5x\,.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:1b“