How and when do you use integration by parts?

Integration by parts is a method of integration used when you are attempting to integrate a function which is the product of two functions. If the two products can be expanded there is usually an easier way to integrate them than integration by parts. For example, x²(x - 4) is easier to integrate when expanded to x³ - 4x².
The general form of the equation for integration by parts is:
∫f(x)g’(x)dx = f(x)g(x) - ∫g(x)f’(x)dx
where f’(x) is the derivative of f(x). It is also commonly seen as:
∫u dv/dx dx = uv - ∫v du/dx dx
where u and v are both function of x.
A good guideline when deciding which function to use as u (or f(x)) is the acronym LIATE:
Logarithmic e.g. ln(x)
Inverse trigonometry e.g. sin^-1(x)
Algebraic e.g. x
Trigonometry e.g. sin(x)
Exponential e.g. e^x.
Step 1:
Split the integrand (function to be integrated) in to its 2 products.
E.g. ∫xln(x)dx can be split in to x and ln(x).
Step 2:
Decide which function should be u and which should be dv/dx.
E.g. x is algebraic, ln is logarithmic. Logarithmic comes before algebraic in LIATE so u = ln(x) and dv/dx = x.
Step 3:
Find du/dx and v by differentiating and integrating u and dv/dx respectively.
E.g. u = ln(x), du/dx = x^-1, dv/dx = x and v = x²/2
Step 4:
Substitute the variables in to the equation for integration by parts.
E.g. ∫xln(x)dx = ln(x)x²/2 - ∫x^-1x²/2 dx = ln(x)x²/2 - ∫x/2 dx.
Step 5:
Evaluate the new integral.
E.g ∫xln(x)dx = ln(x)x²/2 - x²/4 + c = x²/4 (2ln(x) - 1) + c where c is a constant of integration.
Step 5 may require you to perform integration by parts again. Also LIATE does not work in every situation. If it does not work, switch the products used for u and dv/dx and try again.