Lösung 2.3:2b
Aus Online Mathematik Brückenkurs 2
We have a product of two factors in the integrand, so a partial integration does not seem unreasonable. There is nevertheless a problem as regards which factor should be differentiated and which should be integrated. If we choose to differentiate
x3ex2dx=4x4
ex2−
4x4
ex22xdx=41x4ex2−21
x5ex2dx
which just seems to make the integral harder. The solution is instead to substitute
10x3ex2dx=
10x2ex2xdx
we see that the expression
xdx
10x3ex2dx=
10x2ex2xdx=
u=x2du=
x2
dx=2xdx
=
10ueu21du=21
10ueudu
We can then calculate this integral be partial integration, where we differentiate away the factor
10ueudu=21
ueu
10−21
101
eudu=21
1
e1−0
−21
eu
10=21e−21
e1−e0
=21e−21e+21=21