Lösung 2.2:1c
Aus Online Mathematik Brückenkurs 2
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Version vom 10:20, 11. Mär. 2009
With the given variable substitution, \displaystyle u=x^3, we obtain
| \displaystyle du = \bigl(x^3\bigr)'\,dx = 3x^2\,dx |
and because the integral contains \displaystyle x^2 as a factor, we can bundle it together with \displaystyle dx and replace the combination with \displaystyle \tfrac{1}{3}\,du,
| \displaystyle \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{.} |
Thus, the answer is
| \displaystyle \int e^{x^3}x^2\,dx = \frac{1}{3}e^{x^3} + C\,, |
where \displaystyle C is an arbitrary constant.
