MATH 20B
LECTURE #7
INTEGRATION BY PARTS
Integration by Parts is a technique for computing integrals.
INTEGRATION BY PARTS (see p. 400)
∫
∫
′
u(x)v (x)dx = u(x)v(x) −
u′ (x)v(x)dx.
THE ''REVERSE PRODUCT RULE''
Recall the product rule:
THE PRODUCT RULE (SEE P. 143)
If f and g are differentiable functions, then f g is differentiable and
(f g)′ (x) = f (x)g ′ (x) + g(x)f ′ (x).
We see that
∫
(u(x)v ′ (x) + u′ (x)v(x))dx = u(x)v(x) + C.
Rearranging this gives Integration by Parts. Integration by Parts is therefore
a “reverse product rule”.
∫
EXAMPLE. Evaluate x cos(x)dx.
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TIPS
From the book (p. 401):
• Choose u so that u′ is simpler than u itself.
∫
• Choose v so that v = v ′ dx can be evaluated.
My tips:
• For
∫
∫
n
x cos(x)dx,
∫
n
x sin(x)dx, or
xn ex dx, where n is a positive
integer, pick u = xn .
∫
• For
xn (ln(x))m dx, where m is a positive integer, and n ̸= −1, pick
u = (ln(x))m .
∫
• In other cases of
f (x)dx, many times u = f (x) and v ′ = 1 works.
(This tip also mentioned in the book.)
∫
EXAMPLE. Evaluate xex dx.
• Good: u = x and v ′ = ex .
• Bad: u = xex and v ′ = 1.
∫
EXAMPLE. Evaluate ln(x)dx.
INTEGRATING BY PARTS MORE THAN ONCE
Sometimes we repeat integration by parts, and at each step the integral
becomes more simplified.
∫
EXAMPLE. Evaluate x2 cos(x)dx.
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A REDUCTION FORMULA
From the book (see p. 402)
∫
∫
x e dx = x e − n
n x
n x
xn−1 ex dx.
Proof. Do integration by parts with u = xn and v ′ = ex .
∫
EXAMPLE. Evaluate x3 ex dx.
INTEGRATION BY PARTS FOR DEFINITE INTEGRALS
Here is the formula:
∫
a
b
b ∫ b
u(x)v (x)dx = u(x)v(x) −
u′ (x)v(x)dx
′
a
∫
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EXAMPLE. Evaluate
1
a
ln(x)
dx.
x2
SUBSTITUTION AND INTEGRATION BY PARTS
Sometimes we need to do substitution before doing integration by parts.
Often this is the case when part of the integral is sin(f (x)), cos(f (x)), ef (x) ,
or ln(f (x)), where f (x) is something other than x.
∫
EXAMPLE. Evaluate x cos(2x)dx.
∫
EXAMPLE. Evaluate
x5 sin(x2 )dx.
∫
EXAMPLE. Evaluate
√
√
xe
x
dx.
MIDTERM #1
Midterm #1 is Friday, October 19 and covers Sections 5.2-5.6, 6.1-6.3, and
7.1, i.e, everything we have covered up to this point.
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