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}
$
