Editing Math:Integration by parts

Integration by parts is basically just a formula to memorize.
Here's how I write it:

<math>
\int uv = uV-\int u'V
</math>

...where:

<math>
u=u(x), 
v=v(x), 
V=\int v(x) \; \mathrm{d} x, 
u'=\frac{ \mathrm{d} }{ \mathrm{d} x} u(x)
</math>

So written out fully, integration by parts looks like:

<math>
\int u(x)v(x) \;\mathrm{d}x = u(x)\int v(x) \; \mathrm{d}x-\int \left( u'(x) \left( \int v(x) \; \mathrm{d}x) \right) \right) \; \mathrm{d}x
</math>

This formula is very useful for when you have two functions, denoted as <math>u(x)</math> and <math>v(x)</math>, multiplied together into  1 function. 

=== Examples === 
So for instance:
<math>
\int 4x\cos (2-3x) \; \mathrm{d}x 
</math>

can be split up into:

<math>
u(x)=4x, v(x)=\cos (2-3x)
</math>

which then calculating all the necessary stuff:

<math>
u(x) = 4x, v(x) = \cos (2-3x), u'(x) = 4, V(x) = -\frac{\sin(2-3x)}{3} + C
</math>

The first part of the formula,

<math>
uV
= 4x \left( -\frac{\sin(2-3x)}{3} \right) + C
= -\frac{4}{3} x \left( \sin(2-3x) \right) + C
</math>

And the second part,

<math>
\begin{array}{lcl}
\int u'V
= \frac{-4}{3} \int \sin(2-3x) \; \mathrm{d}x
= \frac{-4}{3} \cdot \frac{1}{3} \cdot \sin(2-3x) + C
= \frac{-4}{9} \sin(2-3x) + C
\end{array}
</math>

Now let's finish this up:

<math>
uV - \int u'V
= -\frac{4}{3} x \left( \sin(2-3x) \right) -  \frac{-4}{9} \sin(2-3x) + C
= -\frac{4}{3} x \left( \sin(2-3x) \right) -  \frac{-4}{9} \sin(2-3x) + C
= \fbox{\phantom{ \boxed -\frac{4}{3} x \left( \sin(2-3x) \right) +  \frac{ 4}{9} \sin(2-3x) + C }
</math>