How do you integrate #int 4 e^x ln x dx # using integration by parts?
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.
Integration by parts is:
A helpful acronym for choosing the
L - logarithmic (ex.
I - inverse trigonometric (ex.
A - algebraic (ex.
T - trigonometric (ex.
E - exponential (ex.
In our case, we have a logarithmic (
Applying the parts formula:
We have to do integration by parts a second time for the new integral
Applying what we found:
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