Chapter 7: Integration Techniques, L’Hôpital’s Rule, and Improper Integrals

Section 7.2: Integration by Parts Particularly useful for integrals involving products of algebraic and transcendental functions –Examples: Based on the formula for the derivative of a product:

Section 7.2: Integration by Parts If u' and v' are both continuous, this can be integrated to obtain: This yields Theorem 7.1: Guidelines for Integration by Parts: 1.In order to find u follow the acronym LIPET. 2.Once u is identified the rest of the integral will be dv.

Section 7.2: Integration by Parts

Section 7.3: Trigonometric Integrals This section helps to evaluate integrals of the forms: The following identities can be helpful:

Section 7.3: Trigonometric Integrals Guidelines for evaluating integrals involving sine and cosine (refer to p. 490): 1.If the power of the sine is odd and positive, save one sine factor and convert the rest to cosine. 2.If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sine. 3.If the powers of both sine and cosine are even and nonnegative, make repeated use of the previously-mentioned identities.

Section 7.3: Trigonometric Integrals

Section 7.4: Trigonometric Substitution This helps to solve integrals involving the radicals: Uses the Pythagorean identities:

For integrals involving let u = a sin θ –Then For integrals involving let u = a tan θ –Then For integrals involving let u = a sec θ –Then Section 7.4: Trigonometric Substitution

Section 7.5: Partial Fractions Allows a rational function to be decomposed into simpler rational functions, to which basic integration formulas can be applied Methods of Decomposing N(x)/D(x): 1.Divide if numerator is greater than denominator 2.Factor the denominator 3.Use linear factors 4.Use quadratic factors

Section 7.5: Partial Fractions

Section 7.7: Indeterminate Forms and L’Hôpital’s Rule The forms 0/0 and ∞/ ∞ are called indeterminate because they do not guarantee that a limit exists, if one does exist. L’Hôpital’s Rule helps to solve these types of problems: Therefore, finding the limit of the derivatives allows one to find the limit of an indeterminate form.

Section 7.7: Indeterminate Forms

Section 7.8: Improper Integrals The definition of a proper integral requires that the interval [a, b] be finite. If either or both of the limits of an integral are infinite, or if f has a finite number of infinite discontinuities in the interval [a, b], then the integral is improper.

Section 7.8: Improper Integrals A function f is said to have an infinite discontinuity at c if, from the right or left: or

Section 7.8: Improper Integrals Definition of Improper Integrals with Infinite Integration Limits: 1.If f is continuous on the interval then 2.If f is continuous on the interval then 3.If f is continuous on the interval then

Section 7.8: Improper Integrals