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Naar aanleiding van Kettingregel.

$
\eqalign{
  & \int {ax^3 e^{bx^2 } } dx = \int {\frac{1}
{{2b}}ax^2  \cdot 2bxe^{bx^2 } dx}   \cr
  & Kies\,\,\,f(x) = \frac{1}
{{2b}}ax^2 \,\,\,en\,\,\,g'(x) = 2bxe^{bx^2 }   \cr
  & Zodat\,\,\,f'(x) = \frac{1}
{b}ax\,\,\,en\,\,\,g(x) = e^{bx^2 }   \cr
  & \int {ax^3 e^{bx^2 } } dx = f(x) \cdot g(x) - \int {g(x) \cdot f'(x)}   \cr
  & \int {ax^3 e^{bx^2 } } dx = \frac{1}
{{2b}}ax^2  \cdot e^{bx^2 }  - \int {e^{bx^2 }  \cdot } \frac{1}
{b}ax\,dx  \cr
  & \int {ax^3 e^{bx^2 } } dx = \frac{a}
{{2b}}x^2 e^{bx^2 }  - \int {\frac{a}
{b}xe^{bx^2 } } dx  \cr
  & \int {ax^3 e^{bx^2 } } dx = \frac{a}
{{2b}}x^2 e^{bx^2 }  - \frac{a}
{{2b^2 }}e^{bx^2 }  + C  \cr
  & \int {ax^3 e^{bx^2 } } dx = \frac{a}
{{2b^2 }}e^{bx^2 } \left( {bx^2  - 1} \right) + C \cr}
$

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