Math:Integration by parts

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

[math]\displaystyle{ \int uv = uV-\int u'V }[/math]

...where:

[math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ u(x) }[/math] and [math]\displaystyle{ v(x) }[/math], multiplied together into 1 function.

Examples

So for instance: [math]\displaystyle{ \int 4x\cos (2-3x) \; \mathrm{d}x }[/math]

can be split up into:

[math]\displaystyle{ u(x)=4x, v(x)=\cos (2-3x) }[/math]

which then calculating all the necessary stuff:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ 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 = \boxed { -\frac{4}{3} x \left( \sin(2-3x) \right) + \frac{ 4}{9} \sin(2-3x) + C } }[/math]