MAT 21B Lecture Notes - Lecture 20: Antiderivative, Product Rule
MAT 21B – Lecture 20 – Integration by Parts
• Definition: The definite integral of the function f over the interval [a, b] is the
number
.
• The 3 Parts of MAT 21B
o Part 1: summation formulas and taking their limits.
o Part 2: the fundamental theorem of calculus in computing definite
integrals including their applications.
▪ Applications include: Q from
over [a, b], area, volume, arc
length, surface area, work, force from variable pressure,
moments, and averages.
o Part 3: Other various methods of calculating definite integrals that are
based on part 2 (Fundamental Theorem of Calculus)
• THEOREM – Integration of Parts:
• To keep track of which function is which, make use of this table:
f
g’
f’
g
• Example: Compute the indefinite integral .
Let fx = x ad g’x = . Then applying the table, we obtain:
f(x) = x
g’x =
f’x = 1
g(x) =
Then, .
• Example 2: Evaluate . This tie, ilude the +C whe fidig gx.
Let fx = 3x ad g’x = .
f(x) = 3x
g’x =
f’x = 3
g(x) =
Note: The ostat C1 is writte rather tha just C to distinguish between
the constant present in g(x) and the constant, C2 that will be added at the end of
evaluating a definite integral in general.
This deostrates that addig a +C whe itegratig g’x to get gx is ot
necessary since as seen in the example, the cancel each other.
• Example 3: Compute
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