Integration using substitution and reduction formula?

Question

Use substitution and the reduction formula to find:

$$\int x^4e^{2x}\,\mathrm{d}x$$

Answer 1

Here is an efficient way forward. Let $I(a)$ be the integral defined by

$$\begin{align} I(a)&=\int e^{ax}\,dx\\\\ &=\frac1a e^{ax}+C \end{align}$$

Notice that upon taking a derivative with respect to $a$, we have

$$\begin{align} I'(a)&=\int xe^{ax}\,dx\\\\ &=\left(-\frac1{a^2}+\frac xa\right)e^{ax}+C \end{align}$$

Continuing, we can generate the $n$'th derivative $I^{(n)}(a)$ as

$$\begin{align} I^{(n)}(a)&=\int x^2e^{ax}\,dx\\\\ &=\frac{d^{n}}{da^{n}}\left(\frac1a e^{ax}\right)+C \end{align}$$

To calculate the integral of interest, simply compute

$$\int x^4e^{2x}\,dx=\left.\left(\frac{d^{4}}{da^{4}}\left(\frac1a e^{ax}\right)\right)\right|_{a=2}+C$$

where one can use the General Leibniz Rule to facilitate the differentiation. The rest is left as a simple exercise.

Answer 2

HINT: Integration by parts: $\int f(dg)=fg-\int g(df)$, where $f=x^4$, $df=4x^3dx$, $dg=e^{2x}dx$, and $g=\frac{1}{2}e^{2x}$. Then, we get $$\int x^4e^{2x}=\frac{1}{2}e^{2x}x^4-2\int e^{2x}x^3.$$

Repeat the process with $\int e^{2x}x^3$ until the $x^3$ becomes $1$. Then you can solve and simplify. If you would like me to go on, I can. However I would rather you figure it out yourself.