Download General Strategy for Integration (MS Word)

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.