Lösung 2.2:3a

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

Version vom 12:51, 20. Okt. 2008

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


\displaystyle \int{\left( \begin{matrix} \text{an}\quad \text{expression} \\ \text{in}\quad u \\ \end{matrix} \right)}\centerdot {u}'\,dx,

where \displaystyle u=u\left( x \right) is the actual substitution. In the integral


\displaystyle \int{2x\sin x^{2}\,dx}


we see that the expression \displaystyle x^{2} is the argument for the sine function, as the same time as its derivative \displaystyle \left( x^{2} \right)^{\prime }=2x stands as a factor in front of sine. Therefore, if we set \displaystyle u=x^{2}, the integral, the integral will be of the form


\displaystyle \int{{u}'\sin u\,dx}


Thus, we can use \displaystyle u=x^{2} for the substitution:


\displaystyle \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}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:3a“