Integration by Parts
More Lessons for Calculus
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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.
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) dxLet 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 - 5x
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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