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. ex.

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. 

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