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Lösung 2.2:3f

Aus Online Mathematik Brückenkurs 2

(Unterschied zwischen Versionen)
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Zeile 1: Zeile 1:
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Let's rewrite the integral somewhat:
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Let's rewrite the integral somewhat,
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{{Displayed math||<math>2\sin\sqrt{x}\cdot\frac{1}{2\sqrt{x}}\,\textrm{.}</math>}}
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<math>2\sin \sqrt{x}\centerdot \frac{1}{2\sqrt{x}}</math>
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Here, we see that the factor on the right, <math>1/2\sqrt{x}</math>, is the derivative of the expression <math>\sqrt{x}</math>, which appears in the factor on the left, <math>2\sin \sqrt{x}\,</math>. With the substitution <math>u=\sqrt{x}</math>, the integrand can therefore be written as
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+
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+
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Here, we see that the factor on the right,
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<math>\frac{1}{2\sqrt{x}}</math>
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is the derivative of the expression
+
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<math>\sqrt{x}</math>, which appears in the factor on the left,
+
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<math>2\sin \sqrt{x}</math>
+
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With the substitution
+
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<math>u=\sqrt{x}</math>, the integrand can therefore be written as
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+
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+
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<math>2\sin u\centerdot {u}'</math>
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 +
{{Displayed math||<math>2\sin u\cdot u'</math>}}
and the integral becomes
and the integral becomes
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+
{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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\int \frac{\sin \sqrt{x}}{\sqrt{x}}\,dx
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& \int{\frac{\sin \sqrt{x}}{\sqrt{x}}}\,dx=\left\{ \begin{matrix}
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&= \left\{ \begin{align}
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u=\sqrt{x} \\
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u &= \sqrt{x}\\[5pt]
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du=\left( \sqrt{x} \right)^{\prime }\,dx=\frac{1}{2\sqrt{x}}\,dx \\
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du &= (\sqrt{x}\,)'\,dx = \frac{1}{2\sqrt{x}}\,dx
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\end{matrix}\, \right\} \\
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\end{align}\, \right\}\\[5pt]
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& =2\int{\sin u\,du} \\
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&= 2\int \sin u\,du\\[5pt]
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& =-2\cos u+C \\
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&= -2\cos u+C\\[5pt]
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& =-2\cos \sqrt{x}+C \\
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&= -2\cos\sqrt{x} + C\,\textrm{.}
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\end{align}</math>
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\end{align}</math>}}

Version vom 15:43, 28. Okt. 2008

Let's rewrite the integral somewhat,

2sinx12x.

Here, we see that the factor on the right, 12x , is the derivative of the expression x , which appears in the factor on the left, 2sinx . With the substitution u=x , the integrand can therefore be written as

2sinuu

and the integral becomes

xsinxdx=udu=x=(x)dx=12xdx=2sinudu=−2cosu+C=−2cosx+C.
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:3f“