Chapter 5 Techniques of Integration

Chapter 5 Techniques of Integration 5.1 Integration By Parts If f and g are differentiable functions of x, the Product Rule says Which implies Or, equivalently, as (1) The application of this formula is called integration by parts.

In practice, we usually rewrite (1) by letting This yields the following alternative form for (1)

Example: Use integration by parts to evaluate Solution:

Guidelines for Integration by Parts The main goal in integration by parts is to choose u and dv to obtain a new integral That is easier to evaluate than the original. A strategy that often works is to choose u and dv so that u becomes “simpler” when Differentiated, while leaving a dv that can be readily integrated to obtain v. There is another useful strategy for choosing u and dv that can be applied when the Integrand is a product of two functions from different categories in the list. Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential In this case, you will often be successful if you take u to be the function whose Category occurs earlier in the list and take dv to the rest of the integrand (LIATE). This method does not work all the time, but it works often enough to b e useful.

Example. Evaluate Solution:

Example: Evaluate Solution:

Repeated Integration by Parts Example: Evaluate Solution:

Example: Evaluate Solution:

Integration by Parts for Definite Integrals For definite integrals, the formula corresponding to is

Example: Evaluate Solution:

5.2 Trigonometric Integrals Integrating powers of sine and cosine By applying the integration by parts , we have two reduction formulas

In particular, ……

Integrating Products of Sines and Cosines

Example: Evaluate Solution:

Examples Example: Evaluate

Integrating Powers of Tangent and Secant There are similar reduction formulas to integrate powers of tangent and secant.

In particular, ……

Integrating Products of Tangents and Secents

Examples Example Evaluate

Eliminating Square Roots (optional) Example: Evaluate

Section 5.3 Trigonometric Substitutions We are concerned with integrals that contain expressions of the form In which a is a positive constant. The idea is to make a substitution for x that will eliminate the radical. For example, to eliminate the radical in , we can make the substitution x = a sinx, - /2    /2 Which yields

The most common substitutions are x=a tan  , x=a sin  , and x=asec The most common substitutions are x=a tan  , x=a sin  , and x=asec. They Come from the reference right triangles below.

We want any substitution we use in an integration to be reversible so that we can Change back to the original variable afterward.

Method of Trigonometric Substitution

Example Example 1. Evaluate Solution.

Example Cont.

Examples Example

Example Example. Evaluate , assuming that x5. Solution. Thus,

Example (optional) There are two methods to evaluate the definite integral. Make the substitution in the indefinite integral and then evaluate the definite integral using the x-limits of integration. Make the substitution in the definite integral and convert the x-limits to the corresponding -limits. Example. Evaluate Solution:

8.5 Integrating Rational Functions by Partial Fractions Suppose that P(x) / Q(x) is a proper rational function, by which we mean that the Degree of the numerator is less than the degree of the denominator. There is a Theorem in advanced algebra which states that every proper rational function can Be expressed as a sum Where F1(x), F2(x), …, Fn(x) are rational functions of the form In which the denominators are factors of Q(x). The sum is called the partial fraction Decomposition of P(x)/Q(x), and the terms are called partial fraction.

Finding the Form of a Partial Fraction Decomposition The first step in finding the form of the partial fraction decomposition of a proper Rational function P(x)/Q(x) is to factor Q(x) completely into linear and irreducible Quadratic factors, and then collect all repeated factors so that Q(x) is expressed as A product of distinct factors of the form From these factors we can determine the form of the partial fraction decomposition Using two rules that we will now discuss.

Linear Factors If all of the factors of Q(x) are linear, then the partial fraction decomposition of P(x)/Q(x) can be determined by using the following rule:

Example: Evaluate Solution:

Example: Evaluate Solution:

Quadratic Factors If some of the factors of Q (x) are irreducible quadratics, then the contribution of Those facctors to the partial fraction decomposition of P(x)/Q(x) can be determined From the following rule:

Examples Example. Evaluate Solution.

Example Cont.

5.7 Improper Integrals In this section, we will extend the concept of a definite integral to allow for infinite Intervals of integration and integrands with vertical asymptotes within the interval of Integration. We will call the vertical asymptotes infinite discontinuities, and call the Integrals with infinite intervals of integration or infinite discontinuities within the interval Of integration improper integrals. Here are some examples: Improper integrals with infinite limits of integration: Improper integrals with infinite discontinuities in the interval of integration: Improper integrals with infinite discontinuities and infinite limits of integration: 5/20/2018

Infinite Limits of Integration 5/20/2018

Example: Evaluate Solution: Example: Evaluate Solution: 5/20/2018

Example: For what values of p does the integral converge? Solution 5/20/2018

Example: Evaluate 5/20/2018

Integrands with Vertical Asymptotes 5/20/2018

Example: Evaluate Solution: 5/20/2018

Example: Evaluate Solution: 5/20/2018

Example: Evaluate Solution: 5/20/2018

Tests for Convergence and Divergence When we cannot evaluate an improper integral directly, we try to determine if it converges or diverges. The principal tests for convergence or divergence are the Direct Comparison Test and the Limit Comparison Test.

Examples Example: Decide if converges. Example: Decide if converges