LECTURE 16: INTEGRATION BY PARTS
MINGFENG ZHAO
February 11, 2015
Theorem 1.
Z
p
x dx
=
Z
cos(ax) dx
=
sin(ax) dx
=
sec2 (ax) dx
=
csc2 (ax) dx
Z
eax dx
=
Z
Z
Z
Z
1
dx
− x2
Z
1
dx
x2 + a2
√
a2
=
=
=


xp+1
p+1
+ C,
if p 6= −1
 ln |x| + C,
if p = −1.
1
sin(ax) + C
a
1
− cos(ax) + C
a
1
tan(ax) + C
a
1
− cot(ax) + C
a
1 ax
e +C
a
x
sin−1
+C
a
x
1
tan−1
+C
a
a
Substitution rule for definite integrals
Z
Example 1. Evaluate
Since tan(x) =
Z
tan(x) dx and
sin(x)
, then
cos(x)
Z
tan(x) dx
cot(x) dx.
Z
=
sin(x)
dx
cos(x)
Z
=
=
1
· (−1) du Let u = cos(x), then du = − sin(x)dx
u
Z
1
−
du
u
1
2
MINGFENG ZHAO
=
− ln |u| + C
=
− ln | cos(x)| + C
Since u = cos(x).
By the same computation, we have
Z
cot(x) dx
Z
cos(x)
dx
sin(x)
Z
1
du Let u = sin(x), then du = cos(x)dx
u
=
=
=
ln |u| + C
=
ln | sin(x)| + C
Since u = sin(x).
Theorem 2. Let u = g(x) be differentiable, then
Z
b
Z
0
g(b)
f (g(x))g (x) dx =
f (u) du.
a
Z
Example 2. Compute
π
2
g(a)
sin4 (x) cos(x) dx.
0
Let u = sin(x), then du = cos(x)dx. Then
Z
π
2
sin4 (x) cos(x) dx
Z
1
=
0
u4 du Let u = sin(x)
0
=
1
1 5 u 5 0
=
1
.
5
Integration by parts for indefinite integrals
Recall the product rule, we have
d
[u(x)v(x)] = u0 (x)v(x) + u(x)v 0 (x).
dx
Theorem 3. Let u and v be differentiable, then
Z
Z
u dv = uv −
v du.
That is,
Z
0
uv dx = uv −
Z
vu0 dx.
LECTURE 16: INTEGRATION BY PARTS
3
Z
Example 3. Compute
ln x dx.
Let
1
u = ln x =⇒ du = dx
x
Z
dv = dx =⇒ v = 1 dx = x
Then
Z
Z
ln x dx
=
u dv
Z
= uv − v du
Z
1
= ln x · x − x · dx
x
Z
= x ln x − 1 dx
= x ln −x + C.
Remark 1. The ‘integration by parts’ is the last integration choice to use. dv is the most complicated portion of the
integrand that can be “ easily” integrated, and u is that portion of the integrand whose derivative du is a “ simpler ”
function than u itself. For the choice of u, please follow the ‘ILATE’ order:
Z
Example 4. Evaluate
I
=
Inverse trigonometric function
L
=
Logarithmic function
A
=
Algebraic function
T
=
Trigonometric function
E
=
Exponential function
sin−1 (x) dx.
Let
u = sin−1 (x)
=⇒
du = √
1
dx
1 − x2
Z
dv = dx
=⇒
v=
1 dx = x
4
MINGFENG ZHAO
Then
Z
−1
sin
Z
(x) dx
=
=
=
=
=
=
=
Z
Example 5. Evaluate
u dv
Z
uv − v du
sin
−1
Z
(x) · x −
Z
x· √
1
dx
1 − x2
x
dx
1 − x2
Z
1
1
√ · −
x sin−1 (x) −
du Let w = 1 − x2 , then dw = −2xdx
2
w
Z
1
1
−1
w− 2 dw
x sin (x) +
2
1
1
1
x sin−1 (x) + · 1
w− 2 +1 + C
2 −2 + 1
−1
x sin
√
(x) −
1
=
x sin−1 (x) + w 2 + C
=
x sin−1 (x) + (1 − x2 ) 2 + C
1
Since w = 1 − x2 .
ex cos(x) dx.
Let
u = ex
dv = cos(x)dx
=⇒ du = ex dx
Z
=⇒ v = cos(x) dx = sin(x)
Then
Z
ex cos(x) dx
Z
=
u dv
Z
= uv − v du
= ex · sin(x) −
Z
For
Z
sin(x) · ex dx.
sin(x) · ex dx, use integration by parts again, let
u = ex
dv = sin(x)dx
=⇒ du = ex dx
Z
=⇒ v = sin(x) dx = − cos(x)
LECTURE 16: INTEGRATION BY PARTS
Then
Z
Z
x
e sin(x) dx
=
u dv
Z
= uv − v du
Z
x
= −e · cos(x) +
cos(x) · ex dx.
So we get
Z
x
x
Z
x
e cos(x) dx = e sin(x) + e cos(x) −
ex cos(x) dx.
So we get
Z
Z
Example 6. Evaluate
1 x
1
e sin(x) + ex cos(x) + C.
2
2
ex cos(x) dx =
xex dx.
Let
u = x =⇒ du = dx
Z
x
dv = e dx =⇒ v = ex dx = ex
Then
Z
Z
x
xe dx
=
=
u dv
Z
uv − v du
Z
x
xe − ex dx
=
xe2 − ex + C.
=
Integration by parts for definite integrals
Theorem 4. Let u and v be differentiable, then
Z
b
0
u(x)v (x) d =
a
b
u(x)v(x)|a
Z
−
a
b
v(x)u0 (x) dx.
5
6
MINGFENG ZHAO
Theorem 5. Let R be the region bounded by the graph of a non-negative function f (x), x-axis between a and b, then
the volume of the solid that is generated when the region R is revolved about the x-axis is:
Z b
π[f (x)]2 dx.
a
Example 7. Let R be the region bounded by y = ln x, the x-axis, and the line x = e. Find the volume of the solid
that is generated when the region R is revolved about the x-axis.
By Theorem 5, we have
e
Z
Volume
π(ln x)2 dx
=
1
e
Z
(ln x)2 dx.
= π
1
Let
u = (ln x)2
dv = dx
2
=⇒ du = ln xdx
x
Z
=⇒ v = 1 dx = x.
By Theorem 4, we have
Volume
e
= π (ln x) x1 −
2
Z
= π e−2
Z
e
1
e
2
x · ln x dx
x
ln x dx
1
e
= π [e − 2 (x ln x − x)|1 ]
By Example 3
= π(e − 2e + 2e − 2)
= π(e − 2).
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca