Integration by Parts (solutions, examples, videos)

A-Level Maths: H5-30 Further Integration: Integration by Parts with Definite Integrals
A-Level Maths: H5-30 Further Integration: Integration by Parts with Definite Integrals

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

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.

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

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 = exdx then v = ex

Using the Integration by Parts formula

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

Examples:
∫x5ln(x)dx
∫sin-1(x)dx
∫exsin(x)dx
∫xexdx
∫x2cos(x)dx

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

Examples:
∫xe-xdx
∫lnx – 1 dx
∫x – 5xdx

Integration by Parts – Definite Integral

Evaluate a Indefinite Integral Using Integration by Parts

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

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