Math 175
Notes and Learning Goals
Lesson 3-1a
Integration By Parts 1
• Integration by parts is inspired by the product rule
– Start with product rule: (uv)0 = vu0 + uv 0
– Solve for uv 0 = (uv)0 − vu0
R
R
R
R
– Integrate both sides uv 0 = (uv)0 − vu0 = uv − vu0
– Similar to integration by substitution use du and dv for the derivative terms
Z
Z
Integration By Parts:
udv = uv − vdu
• Use integration by parts to transform the integral
R
f (x)dx as follows
1. Split the integrand, f (x), into two factors: u and dv (include the dx with dv).
2. Find the derivative of u and the antiderivative of dv.
3. Transform the integral using integration by parts:
Z
Z
Z
f (x)dx = udv = uv − vdu
This is just the transformed integral
problem. The first piece, uv is part of the final
R
antiderivative, the second piece vdu is a new integral you need find.
R
4. To find the full antiderivative you still need to find the integral vdu.
– This new integral could be easier or harder than the original, depending on the
choices of u and dv.
– In the case the new integral is easier, find its antiderivative and use it to complete
the original problem.
– Some problems require integration by parts to be use multiple times to find the
original antiderivative.
• Integration by parts with a definite integral:
Z
a
b
b Z b
udv = uv −
vdu
a
a
b
– Use the bounds on the first piece: uv a = u(b)v(b) − u(a)v(a)
– You still have to find the antiderivative of the second piece
the bounds. As above this is a new integral to evaluate.
Rb
a
vdu before you can use
• Integration by parts example videos: Khan Academy or PatrickJMT.
1