Chapter 8: Integration Techniques, L'Hopital's Rule, and Improper Integrals

LHôpitals Rule LHopitals Rule

________- states that under certain conditions, the limit of the quotient f (x)/g (x) is determined by the limit of the quotient of the derivatives f (x)/g (x)

Trigonometric Substitution

________: u= a sin 𝜃𝜃.

Power of Cosine

________ is Even and Nonnegative Find ∫ 𝑠𝑠𝑐𝑐𝑠𝑠 4 𝑥𝑥 𝑑𝑑𝑥𝑥 Solution.

form of ∫ 𝑠𝑠𝑠𝑠𝑠𝑠

When the integral is in the ________ 𝑚𝑚 𝑥𝑥 𝑑𝑑𝑥𝑥, where m is odd and positive, use integration by parts.

Arctangent Rule

The ________ does not apply because the numerator contains a factor of x.

improper fraction

Divide when Improper: When N (x)/D (x) is a(n) ________ (that is, the degree of the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain Where the degree of N1 (x) is less than the degree of D (x)

Note

In Example 1c, some algebra is required before applying any integration rules, and more than one rule is needed to evaluate the resulting integral

Note

Remember that you can separate numerators but not denominators

Substitution

𝑢𝑢 = ln(sin 𝑥𝑥 )

Substitution

𝑢𝑢 = 2𝑥𝑥 Trigonometric Identity

Rewrite as a function of x 8.2 Integration by Parts Integration by Parts

a technique particularly useful for integrands involving products of algebraic and transcendental functions

Theorem

Integration by Parts It is based on the formula for the derivative of a product

Ł 𝑠𝑠 𝑛𝑛𝑥𝑥 cos 𝑏𝑏𝑥𝑥 𝑑𝑑𝑥𝑥 8.3 Trigonometric Integrals Sheila Scott Macintyre (1910-1960)

published her first paper on the asymptotic periods of integral functions in 1935

Objective

To eliminate the radical in the integrand

Trigonometric Substitution

u = a sin 𝜃𝜃

Trigonometric Substitution

Rational Powers

8.5 Partial Fractions Method of Partial Fractions

a procedure for decomposing a rational function into simpler rational functions to which you can apply the basic integration formula

Divide when Improper

When N(x)/D(x) is an improper fraction (that is, the degree of the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain Where the degree of N1(x) is less than the degree of D(x)

Factor Denominator

Completely factor the denominator into factors of the form (𝑝𝑝 𝑥𝑥 + 𝑞𝑞 )𝑚𝑚 and (𝑎𝑎 𝑥𝑥 2 + 𝑏𝑏𝑥𝑥 + 𝑠𝑠 )𝑛𝑛 whereas 𝑎𝑎 𝑥𝑥 2 + 𝑏𝑏𝑥𝑥 + 𝑠𝑠 is irreducible

Linear Factors

For each factor of the form (𝑝𝑝 𝑥𝑥 + 𝑞𝑞 )𝑚𝑚 , the partial fraction decomposition 4

Quadratic Factors

For each factor of the form (𝑎𝑎 𝑥𝑥 2 + 𝑏𝑏𝑥𝑥 + 𝑠𝑠 )𝑛𝑛 , the partial fraction decomposition must include the following sum of n fractions

8.6 Indeterminate Forms and LHôpitals Rule Indeterminate Forms Indeterminate Forms

forms 0/0 and ∞/∞ are called indeterminate because they do not guarantee that the limit exists, nor do they indicate what the limit is if one does exist

Note

All indeterminate forms, however, can be evaluated by algebraic manipulation

LHôpitals Rule LHopitals Rule

states that under certain conditions, the limit of the quotient f(x)/g(x) is determined by the limit of the quotient of the derivatives f(x)/g(x)

8.7 Improper Integrals Improper Integrals with Infinite Limits of Integration Definite Integral

requires that the interval [a, b] be finite