Calc.jjw3.com
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 and ).
2. Simplify the integrand.
I. Some examples:
a. Expand integrand (if you have a product of polynomial factors):
e.g.,[pic]
b. Simplify integrand
i. if you have a single factor in the denominator:
e.g.,[pic]
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.,[pic]
c. Factor, completing the square;
d. Rationalize;
e. Use trigonometric identities ().
3. u Substitution.
Need to know how to differentiate ().
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 [pic] as long as [pic] is in the integrand:
let [pic], then [pic]
i. Special case
(NOTE: the derivative of the inner function is not in the integrand!):
1. [pic]
let [pic], then [pic] or [pic]
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: [pic] (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 [pic], use [pic] where a > 0, [pic];
b. if integrand contains [pic], use [pic] where a > 0, [pic];
c. if integrand contains [pic], use [pic] where a > 0, [pic], [pic].
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 [pic] for polynomials,[pic] and [pic], where degree[pic]< degree[pic]:
a. NOTE: remember to use:
i. [pic] for n > 1.
ii. [pic] for irreducible quadratic, [pic].
iii. [pic] by completing the square.
II. Integrand is [pic] for polynomials [pic],[pic] where degree [pic] ≥ degree [pic]:
a. use polynomial division, then use method of partial fractions.
7. Trigonometric Integrals.
I. [pic], n ≥ 1 is odd, m ≥ 0 is even
a. Convert all but one [pic] into [pic] using [pic]
b. let [pic]
II. [pic], n ≥ 0 is even, m ≥ 1 is odd
a. Convert all but one [pic] into [pic] using [pic]
b. let [pic]
III. [pic], n ≥ 1 is odd and m ≥ 1 is odd
a. Use one of the above methods.
IV. [pic], n ≥ 0 is even, m ≥ 0 is even
a. Start with [pic] or [pic] repeatedly to reduce the
integrand into [pic], where b is even.
b. NOTE: You may need to use these double-angle formulas multiple times.
V. [pic], [pic], or [pic]
a. Start with sum or difference formulas.
VI. [pic], n > 2 is even
a. Convert all but two [pic] into [pic] using [pic]
b. let [pic]
VII. [pic], n ≥ 1 is odd, m ≥ 1 is odd
a. Convert all but one [pic] into [pic] using [pic]
b. let [pic]
VIII. [pic], n = 0, m ≥ 2 is even
a. Convert all but one [pic] into [pic] using [pic], expand, repeat
as necessary
IX. [pic], n ≥ 1, m = 0 is odd
a. Use integration by parts
IX. [pic] is analogous to integrals involving sec(x) and tan(x).
X. All other cases, try to convert into [pic] and [pic], simplify.
8. Radicals (integrals involving [pic])
I. try the substitution [pic] 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.
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