# 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, x^{2}(x - 4) is easier to integrate when expanded to x^{3} - 4x^{2}.

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**:

**L**ogarithmic e.g. ln(x)

**I**nverse trigonometry e.g. sin^{-1}(x)

**A**lgebraic e.g. x

**T**rigonometry e.g. sin(x)

**E**xponential 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 = x^{2}/2

**Step 4:**

Substitute the variables in to the equation for integration by parts.

E.g. ∫xln(x)dx = ln(x)x^{2}/2 - ∫x^{-1}x^{2}/2 dx = ln(x)x^{2}/2 - ∫x/2 dx.

**Step 5:**

Evaluate the new integral.

E.g ∫xln(x)dx = ln(x)x^{2}/2 - x^{2}/4 + c = x^{2}/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.