Integration by Parts

In these lessons we learn how to work out integrals using integration by parts.

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Related Pages
Calculus: Integration
Calculus: Derivatives
Calculus Lessons

The following figures give the formula for Integration by Parts and how to choose u and dv. Scroll down the page for more examples and solutions.

Integration by Parts

How to derive the rule for Integration by Parts from the Product Rule for differentiation?

The Product Rule states that if f and g are differentiable functions, then


Integrating both sides of the equation, we get


We can use the following notation to make the formula easier to remember.

Let u = f(x) then du = f‘(x) dx

Let v = g(x) then dv = g‘(x) dx

The formula for Integration by Parts is then

Integration by Parts


Let u = x then du = dx

Let dv = sin xdx then v = –cos x

Using the Integration by Parts formula

Integration by Parts Example





Let u = x2 then du = 2x dx

Let dv = exdx then v = ex

Using the Integration by Parts formula

integration by parts

We use integration by parts a second time to evaluate

Let u = x the du = dx

Let dv = ex dx then v = ex


Substituting into equation 1, we get


Integration by parts - choosing u and dv
How to use the LIATE mnemonic for choosing u and dv in integration by parts?
Let u be the first thing in this list and dv be everything else
Logarithmic functions
Inverse Trig functions
Algebraic functions
Trig functions
Exponential functions


Integration by Parts
3 complete examples are shown of finding an antiderivative using integration by parts.

∫lnx - 1 dx
∫x - 5xdx

Integration by Parts - Definite Integral

Evaluate a Indefinite Integral Using Integration by Parts

Use integration by parts to evaluate the integral:
∫ln(3r + 8)dr

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