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Lösung 2.2:1c

Aus Online Mathematik Brückenkurs 2

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With the given variable substitution, <math>u=x^3</math>, we obtain
With the given variable substitution, <math>u=x^3</math>, we obtain
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{{Displayed math||<math>du = \bigl(x^3\bigr)'\,dx = 3x^2\,dx</math>}}
+
{{Abgesetzte Formel||<math>du = \bigl(x^3\bigr)'\,dx = 3x^2\,dx</math>}}
and because the integral contains <math>x^2</math> as a factor, we can bundle it together with <math>dx</math> and replace the combination with <math>\tfrac{1}{3}\,du</math>,
and because the integral contains <math>x^2</math> as a factor, we can bundle it together with <math>dx</math> and replace the combination with <math>\tfrac{1}{3}\,du</math>,
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{{Displayed math||<math>\int e^{x^3}x^2\,dx = \bigl\{\,u=x^3\,\bigr\} = \int e^u\tfrac{1}{3}\,du = \frac{1}{3}e^u + C\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>\int e^{x^3}x^2\,dx = \bigl\{\,u=x^3\,\bigr\} = \int e^u\tfrac{1}{3}\,du = \frac{1}{3}e^u + C\,\textrm{.}</math>}}
Thus, the answer is
Thus, the answer is
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{{Displayed math||<math>\int e^{x^3}x^2\,dx = \frac{1}{3}e^{x^3} + C\,,</math>}}
+
{{Abgesetzte Formel||<math>\int e^{x^3}x^2\,dx = \frac{1}{3}e^{x^3} + C\,,</math>}}
where <math>C</math> is an arbitrary constant.
where <math>C</math> is an arbitrary constant.

Version vom 13:00, 10. Mär. 2009

With the given variable substitution, u=x3, we obtain

du=x3dx=3x2dx 

and because the integral contains x2 as a factor, we can bundle it together with dx and replace the combination with 31du,

ex3x2dx=u=x3=eu31du=31eu+C. 

Thus, the answer is

ex3x2dx=31ex3+C 

where C is an arbitrary constant.

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