Integration by Parts
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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