General Strategy for Integration
for Calculus II
As you gain experience from practice, you will more quickly recognize which method will work best for
a particular integral. You are not expected to memorize these methods, but you are expected to know the
methods. Keep in mind for some integrals, more than one method may work - the answers that result
from different methods may look different but will be equivalent.
1. Use known formulas from your knowledge of derivatives (p. 318; compare
http://calc.jjw3.com/math2431/diffFormulas.doc and
http://calc.jjw3.com/math2431/basicIntegrationFormulas.doc).
2. Simplify the integrand.
I. Some examples:
a. Expand integrand (if you have a product of polynomial factors):
e.g.,   x  3  x 2  5  dx    x 3  3x 2  5 x  15  dx
b. Simplify integrand
i. if you have a single factor in the denominator:
e.g., 
 x 2 3x 2 
x 2  3x  2
2

dx       dx    x  3   dx
x
x x
x

 x
ii. if you have a polynomial in the denominator that has one of the forms
of known integral formulas or has one of the forms in the table of integrals:
e.g., 
x3
x
3
dx   2
dx   2
dx
2
x 5
x 5
x 5
c. Factor, completing the square;
d. Rationalize;
e. Use trigonometric identities (http://calc.jjw3.com/math2432/TrigFormulas.doc).
3. u Substitution.
Need to know how to differentiate (http://calc.jjw3.com/math2431/diffFormulas.doc).
Substitution is preferable (easier/faster) to trigonometric substitution and partial fractions.
I. Possible substitutions (you need to have u and u′ both in integrand):
a. Composition of functions of the form f  g  x   as long as g   x  is in the integrand:
let u  g  x  , then du  g   x  dx
i. Special case
(NOTE: the derivative of the inner function is not in the integrand!):
1.
 f  x  dx
let u  x , then du 
1
2 x
dx or dx  2udu
b. let u = denominator function or factor in denominator;
c. let u = factor in numerator when u′ is included with denominator.
II. Check the following:
a. the entire integrand is in terms of the new variable, u (do not forget the du);
i. NOTE: you may need to use the let statement for additional substitution;
b. if necessary, the limits of integration are converted into values of u.
4. Integration by parts:  UdV  UV   VdU (You need to know how to integrate dV and VdU).
Table method shown in class is typically faster.
a. possibly used when the integrand contains a product of any two or more of the following
types of functions:
polynomial, trigonometric, exponential, logarithmic, and inverse.
I. Some suggestions (use these suggestions cautiously):
a. let U = log function
b. let U = polynomial function
c. let U = inverse function; dV = dx
II. If one of the factors has the form of the derivative of the other factor, then you may
want to try substitution.
5. Trigonometric substitution
I. Suggested substitutions:
a. if integrand contains  a 2  x2  2 , use x  a sin   where a > 0, 
n

b. if integrand contains  a 2  x2  2 , use x  a tan   where a > 0, 
n
2

2
 
 
c. if integrand contains  x2  a 2  2 , use x  a sec   where a > 0, 0   
n
  
3
.
2

2

2

2
;
;
,
II. Procedure:
a. substitute x and dx into the original integrand;
b. simplify the integrand;
c. integrate;
d. use right triangle to convert back into terms of x.
6. Partial fractions.
I. Integrand
P  x
for polynomials, P  x  and Q  x  , where degree P  x  < degree Q  x  :
Q  x
a. NOTE: remember to use:
i.
ii.
iii.
II. Integrand is
P( x)
 bx  c 
n

A
B
M

  
for n > 1.
2
n
bx  c  bx  c 
 bx  c 
P( x)
Ax  B
 2
for irreducible quadratic, bx 2  d .
2
bx  d bx  d
P( x)
P( x)
by completing the square.

2
bx  cx  d
c  4bd  c 2

b x   
2b 
4b

2
P  x
for polynomials P  x  , Q  x  where degree P  x  ≥ degree Q  x  :
Q  x
a. use polynomial division, then use method of partial fractions.
7. Trigonometric Integrals.
I.  cos n  x  sin m  x  dx , n ≥ 1 is odd, m ≥ 0 is even
a. Convert all but one cos  x  into sin  x  using cos2  x   1  sin 2  x 
b. let u  sin  x 
II.  cos n  x  sin m  x  dx , n ≥ 0 is even, m ≥ 1 is odd
a. Convert all but one sin  x  into cos  x  using sin 2  x   1  cos2  x 
b. let u  cos  x 
III.  cos n  x  sin m  x  dx , n ≥ 1 is odd and m ≥ 1 is odd
a. Use one of the above methods.
IV.  cos n  x  sin m  x  dx , n ≥ 0 is even, m ≥ 0 is even
a. Start with cos 2  x  
1  cos  2 x 
or sin 2  x  
2
integrand into cos  bx  , where b is even.
1  cos  2 x 
2
repeatedly to reduce the
b. NOTE: You may need to use these double-angle formulas multiple times.
V.  cos  nx  sin  mx  dx ,  cos  nx  cos  mx  dx , or  sin  nx  sin  mx  dx
a. Start with sum or difference formulas.
VI.  sec n  x  tan m  x  dx , n > 2 is even
a. Convert all but two sec  x  into tan  x  using sec2  x   1  tan 2  x 
b. let u  tan  x 
VII.  sec n  x  tan m  x  dx , n ≥ 1 is odd, m ≥ 1 is odd
a. Convert all but one tan  x  into sec  x  using tan 2  x   sec2  x   1
b. let u  sec  x 
VIII.  sec n  x  tan m  x  dx , n = 0, m ≥ 2 is even
a. Convert all but one tan 2  x  into sec  x  using tan 2  x   sec2  x   1, expand, repeat
as necessary
IX.  sec n  x  tan m  x  dx , n ≥ 1, m = 0 is odd
a. Use integration by parts
IX.  csc n  x  cot m  x  dx is analogous to integrals involving sec(x) and tan(x).
X. All other cases, try to convert into sin  x  and cos  x  , simplify.
8. Radicals (integrals involving
n
ax  b )
I. try the substitution u n  ax  b and implicitly differentiate to find du).
9. Use multiple methods.
10. Use Table of integrals.
11. If at first you don’t succeed, try again. Sometimes a combination of the above methods may be
required. Most important of all, practice.