How do you integrate int 4 e^x ln x dx  using integration by parts?

Apr 4, 2016

You can try, but you'll find that this integral can produce an infinity of integrals if you keep applying integration by parts over and over.

Explanation:

Integration by parts is:
$\int u \mathrm{dv} = u v - \int v \mathrm{du}$
A helpful acronym for choosing the $u$ of IbP is LIATE :
L - logarithmic (ex. $\ln x$, ${\log}_{3} x$)
I - inverse trigonometric (ex. ${\tan}^{- 1} x , \arcsin \left(x\right)$)
A - algebraic (ex. ${x}^{2}$, $\frac{1}{x}$)
T - trigonometric (ex. $\sin x$, $\cos x$)
E - exponential (ex. ${e}^{x}$, ${2}^{x}$)

In our case, we have a logarithmic ($\ln x$) and exponential ($4 {e}^{x}$) function. In our LIATE list, logarithmic functions come first, so our $u$ will be $\ln x$. That means everything else, namely $4 {e}^{x} \mathrm{dx}$ will be $\mathrm{dv}$:
$u = \ln x \to \frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{x} \to \mathrm{du} = \frac{1}{x} \mathrm{dx}$
$\mathrm{dv} = 4 {e}^{x} \mathrm{dx} \to \int \mathrm{dv} = \int 4 {e}^{x} \mathrm{dx} \to v = 4 {e}^{x}$

Applying the parts formula:
$\int 4 {e}^{x} \ln x \mathrm{dx} = \left(\ln x\right) \left(4 {e}^{x}\right) - \int 4 {e}^{x} \frac{1}{x} \mathrm{dx}$
$\textcolor{w h i t e}{X X} = 4 {e}^{x} \ln x - 4 \int {e}^{x} \frac{1}{x} \mathrm{dx}$

We have to do integration by parts a second time for the new integral $4 \int {e}^{x} \frac{1}{x} \mathrm{dx}$:
$u = \frac{1}{x} \to \frac{\mathrm{du}}{\mathrm{dx}} = - \frac{1}{x} ^ 2 \to \mathrm{du} = - \frac{1}{x} ^ 2 \mathrm{dx}$
$\mathrm{dv} = {e}^{x} \mathrm{dx} \to \int \mathrm{dv} = \int {e}^{x} \mathrm{dx} \to v = {e}^{x}$

Applying what we found:
$\int 4 {e}^{x} \ln x \mathrm{dx} = 4 {e}^{x} \ln x - 4 \left(\frac{1}{x} {e}^{x} - \int {e}^{x} \left(- \frac{1}{x} ^ 2 \mathrm{dx}\right)\right)$
$\textcolor{w h i t e}{X X} = 4 {e}^{x} \ln x - 4 \left({e}^{x} / x + \int {e}^{x} \frac{1}{x} ^ 2 \mathrm{dx}\right)$

Ahhhhh! We have to use another integration by parts! But before you do, notice what's going on. Each time we integrate by parts, we end up with something like ${e}^{x} \frac{1}{x}$ in a new integral. This is, in fact, a pattern. If we keep integrating by parts, we'll get one integral after another, in the form $\int {\left(- 1\right)}^{n} \left({e}^{x} / {x}^{n}\right) \mathrm{dx}$ - and it will never stop. So this integral continues on forever with no solution.