# Calculus

### Presentations text content in Calculus

Calculus

Chapter 7 Day 1

Slide2Basic Integration Rules

Fitting Integrands to Basic Rules

Slide3Fitting 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.

Slide4Comparison of Three Similar Integrals

A. B.C.

Slide5

A.

Use the Arctangent Rule and let and .

Slide6

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

Slide7

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.

Slide8

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.

Slide9

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.

Slide10Using 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.

Slide11Helpful 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

Slide12Integration by Parts

Integration by parts- Tabular Method

Slide13Integration 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.

Slide14Integration by Parts

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

Slide15

Integration by Parts

If and are functions of and have continuous derivatives, then

Slide16

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.

Slide17

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

Slide18

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.

Slide19

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.

Slide20

Common Integrals Using Integration by Parts

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

Slide21Assignment 7-1

Slide22Trigonometric Integrals

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

Slide23Integrals Involving Powers of Sine and Cosine

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

Slide24Integrals Involving Powers of Secant and Tangent

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

Slide25Integrals Involving Sine-Cosine Products with Different Angles

Slide26

Trigonometric Substitution

trigonometric Substitution- Applications

Slide27Trigonometric Substitution

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

Slide28

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 .

Slide29

Special Integration Formulas ()

1. 2. 3.

Slide30

Assignment 7-2

Slide31Slide32

Slide33

Slide34

## Calculus

Download Presentation - The PPT/PDF document "Calculus" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.