Integration of $x^3 e^x$

Question

I am beginner in calculus and I am struggling with this integral:

$$\int x^{3}e^{x}\mathrm{d}x$$

If anyone could give me some hints, any help will be appreciated.

Answer 1

Either integrate by parts, or write that, when $P(x)$ is a polynomial,

$$\int P(x)e^x \mathrm{d}x=Q(x)e^x+C$$

Where $Q$ is a polynomial and $C$ a constant.

You have thus, differentiating:

$$(Q(x)+Q'(x))e^x=P(x)e^x$$

Or $Q+Q'=P$.

Hence, $Q$ has same degree as $P$, and same coefficient of highest degree, so $Q(x)=x^3+ax^2+bx+c$, and $Q'(x)=3x^2+2ax+b$, hence

$$x^3=x^3+(a+3)x^2+(b+2a)x+b+c$$

Thus $a+3=0$, so $a=-3$, then $b+2a=0$, so $b=-2a=6$, and $c=-b=-6$.

You have thus

$$\int x^3e^x \mathrm{d}x=(x^3-3x^2+6x-6)e^x+C$$

Answer 2

Here 3 Integrations by parts will help, as $e^x$ will always reproduce itself and $x^3$ is constant after 3 derivatives.