Lösung 2.2:3b

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If we are to succeed in simplifying the integral with a substitution, we must find an expression <math>u = u(x)</math> so that the integral can be written as
If we are to succeed in simplifying the integral with a substitution, we must find an expression <math>u = u(x)</math> so that the integral can be written as
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{{Displayed math||<math>\int \left(\begin{matrix}
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{{Abgesetzte Formel||<math>\int \left(\begin{matrix}
\text{something}\\
\text{something}\\
\text{in u}
\text{in u}
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As our integral is written,
As our integral is written,
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{{Displayed math||<math>\int\sin x\cos x\,dx</math>}}
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{{Abgesetzte Formel||<math>\int\sin x\cos x\,dx</math>}}
we see that the second factor <math>\cos x</math> is a derivative of the first factor, <math>\sin x</math>. If <math>u=\sin x</math>, the integral can thus be written as
we see that the second factor <math>\cos x</math> is a derivative of the first factor, <math>\sin x</math>. If <math>u=\sin x</math>, the integral can thus be written as
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{{Displayed math||<math>\int u\cdot u'\,dx</math>}}
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{{Abgesetzte Formel||<math>\int u\cdot u'\,dx</math>}}
and this makes <math>u=\sin x</math> an appropriate substitution,
and this makes <math>u=\sin x</math> an appropriate substitution,
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
\int \sin x\cos x\,dx
\int \sin x\cos x\,dx
&= \left\{ \begin{align}
&= \left\{ \begin{align}

Version vom 13:01, 10. Mär. 2009

If we are to succeed in simplifying the integral with a substitution, we must find an expression \displaystyle u = u(x) so that the integral can be written as

\displaystyle \int \left(\begin{matrix}

\text{something}\\ \text{in u} \end{matrix}\right)\cdot {u}'\,dx\,\textrm{.}

As our integral is written,

\displaystyle \int\sin x\cos x\,dx

we see that the second factor \displaystyle \cos x is a derivative of the first factor, \displaystyle \sin x. If \displaystyle u=\sin x, the integral can thus be written as

\displaystyle \int u\cdot u'\,dx

and this makes \displaystyle u=\sin x an appropriate substitution,

\displaystyle \begin{align}

\int \sin x\cos x\,dx &= \left\{ \begin{align} u &= \sin x\\[5pt] du &= (\sin x)'\,dx = \cos x\,dx \end{align} \right\}\\[5pt] &= \int u\,du\\[5pt] &= \frac{1}{2}u^{2} + C\\[5pt] &= \frac{1}{2}\sin^2\!x + C\,\textrm{.} \end{align}

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