Integration by parts
Integration by parts is the product rule backwards. The key is the choice of u and v': the new integral has to be simpler.
Rules
- : the part that gets simpler when differentiated (polynomials, )
- : the part you can integrate easily (, , )
- Trick for :
Your turn
Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).
Task 1 of 3Integration by parts
Task
Find an antiderivative:
(with or without )
Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.
Worked example
Find an antiderivative:
Worked solution
- 1.Integration by parts with , : .
- 2.
- 3.
Typical mistake: Forgetting the minus when integrating : .
Common questions
How do I choose u and v'?
Rule of thumb LIATE: logarithms, inverse functions, algebraic (polynomials), trigonometric, exponential. Whatever comes first becomes u.