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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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