VIII. PRINCIPLES OF INTEGRAL EVALUATION
8.1 Integration by Parts
 f ( x) g ' ( x)dx 
1. Formula for integration by parts:
 udv  uv   vdu
f ( x) g ( x)   g ( x) f ' ( x)dx , or
for u  f (x) , du  f ' ( x)dx , v  g (x ) , dv  g ' ( x)dx
Key points for using integration by parts: (a) v(x) is easy to be derived. (b) The evaluation
of
2. Integration by parts for definite integrals:
 vdu
b
b
a
a
 udv  uv
8-1
is easier than
b
  vdu
a
 udv .
8.2 Trigonometric integrals
1. Integrating powers of sine and cosine:
 sin
 cos
n
1
n 1
xdx   sin n 1 x cos x 
sin n  2 xdx
n
n 
n
xdx 
1
n 1
cos n 1 x sin x 
cos n  2 xdx
n
n 
2. Integrating products of sines and cosines:  sin m x cos n xdx , m, n are positive integers.
(a) If m and n are even: Use the identities sin 2 x 
1  cos 2 x
1  cos 2 x
and cos 2 x 
to
2
2
reduce the powers on sinx and cosx.
(b) If m is odd: Split off a factor of sinx. Applying the identity sin 2 x  1  cos 2 x . Make
the substitution u  cos x .
(c) If n is odd: Split off a factor of cosx. Applying the identity cos 2 x  1  sin 2 x . Make
the substitution u  sin x .
3. Integrating powers of tangent and secant:
n
 tan xdx 
n
 sec xdx 
tan n 1 x
  tan n 2 xdx
n 1
sec n 2 x tan x n  2

sec n 2 xdx

n 1
n 1
4. Integrating products of tangent and secants:  tan m x sec n xdx , m and n are positive integers.
(a) If n is even: Split off a factor of sec 2 x . Apply the identity sec 2 x  tan 2 x  1 . Make
the substitution u=tanx.
(b) If m is odd: Split off a factor of secxtanx. Apply the identity tan 2 x  sec 2 x  1 . Make
the substitution u=secx.
(c) If m is even and n is odd: Use the identities tan 2 x  sec 2 x  1 to reduce the integrand
to powers of secx alone. Then use the reduction formula for
powers of secx.
8.3 Trigonometric Substitutions
8-2
1. The method of trigonometric substitution
Expression in
Substitution
the integrand
Restriction on 
a2  x2
x  a sin 

a2  x2
x  a tan 


2

2
0  
x a
2
2
x  a sec

2
 
 

2

2

2
Simplification
a 2  x 2  a 2  a 2 sin 2   a 2 cos 2 
a 2  x 2  a 2  a 2 tan 2   a 2 sec 2 
, if x  a
x 2  a 2  a 2 sec 2   a 2  a 2 tan 2 
    , if x   a
2. Integrals involving ax 2  bx  c
8.4 Integrating Rational Functions by Partial Fractions
1. Linear factor rule: For each factor of the form (ax  b) m , the partial fraction decomposition
Am
A1
A2
contains the following sum of m partial fractions:




ax  b (ax  b) 2
(ax  b) m
where A1 , A2 ,  , Am are constant to be determined.
2. Quadratic factor rule: For each factor of the form (ax 2  bx  c) m , the partial fraction
decomposition
contains
the
following
sum
Am x  Bm
A1 x  B1
A2 x  B2


2
2
2
ax  bx  c (ax  bx  c)
(ax 2  bx  c) m
of
where
m
partial
fractions:
A1 , A2 ,  , Am
,
B1 , B2 ,  , Bm are constant to be determined.
3. Improper rational function can be integrated by performing a long division and expressing
the function as the quotient plus the remainder over the divisor.
8-3
8.5 Using Table of Integrals and Computer Algebra System
1. Matches requiring substitutions
2. Matches requiring reduction formulas
3. Matches requiring special substitutions
8-4
8-5
8.6 Improper Integrals
1. Improper integrals: the integrals with infinite intervals of integration or infinite
discontinuities within the interval of integration
2. Integrals over infinite intervals
8-6
(a)
The


a
improper
f ( x)dx  lim
integral

l
l  a
of
f
over
the
interval
[a, ) is
defined
as
f ( x)dx . In the case where the limit exists, the improper integral is
said to converge, and the limit is defined to the value of the integral. In the case where
the limit does not exist, the improper integral is said to diverge, and it is not assigned a
value.
(b) The improper integral of

b

f ( x)dx  lim

b
l  l
f over the interval
(, b]
is defined as
f ( x)dx . The integral is said to converge if the limit exists and
diverge if it does not.
(c) The improper integral of f over the interval



c


c
f ( x)dx   f ( x)dx  
(, )
is defined as
f ( x)dx where c is any real number. The improper
integral is said to converge if both terms converge and diverge if either term diverges.
3. Integrals whose integrands have infinite discontinuities:
(a)
If f is continuous on the interval [a, b], except for an infinite discontinuity at b, then
the improper integral of f over the interval [a, b] is defined as
8-7

b
a
l
f ( x)dx  lim  f ( x)dx .
l b
a
In the case where the limit exists, the improper integral is
said to converge, and the limit is defined to be the value of the integral. In the case
where the limit does not exist, the improper integral is said to diverge, and it is not
assigned a value.
(b)
If f is continuous on the interval [a, b], except for an infinite discontinuity at a, then
the improper integral of f over the interval [a, b] is defined as

b
a
b
f ( x)dx  lim  f ( x)dx . The integral is said to converge if the limit exists and
l a
l
diverge if it does not.
(c)
If f is continuous on the interval [a, b], except for an infinite discontinuity at a point c
in (a, b), then the improper integral of f over the interval [a, b] is defined as
8-8

b
a
c
b
a
c
f ( x)dx   f ( x)dx   f ( x)dx . The improper integral is said to converge if both
terms converge and diverge if either term diverges.
8-9