Calculus

Calculus

Chapter 7 Day 1

Basic Integration Rules

Fitting Integrands to Basic Rules

Fitting Integrands to Basic Rules

So far we have dealt with only basic integration rules. But what happens when our integral doesn’t fit into one of those categories? What then?

We have a few techniques we are going to learn that will help us handle these situations.

Comparison of Three Similar Integrals

A. B.C.

A.

Use the Arctangent Rule and let and .

B.

Here the Arctangent Rule does not apply because the numerator contains a factor of . Consider the Log Rule and let . Then and you have

C.

Because the degree of the numerator is equal to the degree of the denominator, you should first use division to rewrite the improper rational function as the sum of a polynomial and a proper rational function.

Using Two Basic Rules to Solve a Single Integral

Evaluate Begin by writing the integral as the sum of two integrals. Then apply the Power Rule and the Arcsine Rule as follows.

A Few More “Fun” Examples

Turn to page 477 in your text book and take a look at example 4 and example 5 for a couple different ways of re-writing integrals to make them “fit” conventional basic rules.

Using Trig Identities

You may also use trigonometric identities for substitutions, so that the integral is easy to solve.

These trig identities can be found on the back cover of your text book.

Helpful Hints

Take a look at page 478-

The gray box on this page provides a helpful set of steps for make the integral “fit” into basic rules.

This is something helpful to remember for the future

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Integration by Parts

Integration by parts- Tabular Method

Integration by Parts

The technique of integration by parts can be applied to a wide variety of functions and is particularly useful for integrands involving products of algebraic and transcendental functions.

Integration by Parts

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

Integration by Parts

If and are functions of and have continuous derivatives, then

Helpful hints (from your book- page 481)

1. Try letting be the most complicated portion of the integrand that fits a basic integration rule. Then will be the remaining factor(s) of the integrand.2. Try letting be the portion of the integrand whose derivative is a function simpler than . Then will be the remaining factor(s) of the integrand.

Integration by Parts-Evaluate:

To apply integration by parts, you need to write the integral in the form . There are several ways to do this:A. where and B. where and C. where and D. where and

Integration by Parts- Continued

The Helpful Hints suggest choosing the first option because the derivative of the is a simpler than , and is the most complicated portion of the integrand that fits a basic integration formula.

Integration by Parts- Continued

To check if you did it correctly, you can always find the derivative of the result and see if you get the original integrand.

Common Integrals Using Integration by Parts

See page 486 for a summary list of Common Integrals Using Integration by Parts

Assignment 7-1

Trigonometric Integrals

Integrals Involving Powers of Sine and Cosine- Integrals Involving Powers of Secant and Tangent- Integrals Involving Sine-Cosine Products with Different Angles

Integrals Involving Powers of Sine and Cosine

See page 490 for a list of guidelines for Evaluating Integrals Involving Sine and Cosine

Integrals Involving Powers of Secant and Tangent

See page 493 for a list of guidelines for Evaluating Integrals Involving Secant and Tangent

Integrals Involving Sine-Cosine Products with Different Angles

Trigonometric Substitution

trigonometric Substitution- Applications

Trigonometric Substitution

The objective of trigonometric substitution is to eliminate the radical in the integrand. You do this with the Pythagorean Identities.

Trigonometric Substitution

1. For integrals involving let . Then where .2. For integrals involving let . Then where .3. For integrals involving let . Then where or Use the positive value if and the negative value if .

Special Integration Formulas ()

1. 2. 3.

Assignment 7-2