Integration by Parts

We can 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

Example:

Evaluate

Solution:

Let u = x then du = dx

Let dv = sin xdx then v = –cos x

Using the Integration by Parts formula

Example:

Evaluate

Solution:

Example:

Evaluate

Let u = x² then du = 2x dx

Let dv = e^xdx then v = e^x

Using the Integration by Parts formula

We use integration by parts a second time to evaluate

Let u = x the du = dx

Let dv = e^x dx then v = e^x

Substituting into equation 1, we get

Videos

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

Integration by Parts - Definite Integral

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