Question

Integrate the function w..r. to x : $x^{3} \cdotp {e^{x}}^{2}$

Answer 1

Solution:

By substitution we can convert it into integrable form
Let
$x^{2} = t\\\\ 2x dx = dt\\\\x dx = \frac{dt}{2}$

$\int x^{3}e^{x^{2}} dx = \int x^{2}\:e^{x^{2}}\:x\:dx\\\\ =\frac{1}{2}\int t\:e^{t} \:dt$

This is the form which can be solved by parts

$\int t\:e^{t} dt=t \int e^{t} dt-\int[\frac{dt}{dt}\int e^{t} dt]dt\\\\=t\:e^t-\int e^{t} dt\\\\\frac{1}{2}\int t\:e^{t} dt=\frac{1}{2}t\:e^t-\frac{1}{2}e^{t}\\\\=\frac{1}{2}e^{t}(t-1)+C$

Now undo substitution

$\int x^{3}e^{x^{2}} dx=\frac{1}{2}e^{x^{2}}(x^{2}-1)+C$