Integrating Rational Functions

Slides:

Advertisements

Similar presentations

6.2 Antidifferentiation by Substitution

Advertisements

6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002.

Integrals of Exponential and Logarithmic Functions.

11 The student will learn about: §4.3 Integration by Substitution. integration by substitution. differentials, and.

TECHNIQUES OF INTEGRATION

Integration By Parts (9/10/08) Whereas substitution techniques tries (if possible) to reverse the chain rule, “integration by parts” tries to reverse the.

Antidifferentiation TS: Making decisions after reflection and review.

Integration by Parts Objective: To integrate problems without a u-substitution.

Computer simulation of the Indonesian tsunami of December 26, 2004 (8 minutes after the earthquake), created using models of wave motion based on advanced.

Warm-up: Evaluate the integrals. 1) 2). Warm-up: Evaluate the integrals. 1) 2)

The Natural Logarithmic Function

1 Numerical Integration Section Why Numerical Integration? Let’s say we want to evaluate the following definite integral:

Indefinite integrals Definition: if f(x) be any differentiable function of such that d/dx f( x ) = f(x)Is called an anti-derivative or an indefinite integral.

Formal Definition of Antiderivative and Indefinite Integral Lesson 5-3.

5.4 The Fundamental Theorem. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in,

Ch8: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies in a.

4-5 INTEGRATION BY SUBSTITUTION MS. BATTAGLIA – AP CALCULUS.

Chapter 7 – Techniques of Integration

5.c – The Fundamental Theorem of Calculus and Definite Integrals.

6.2 Integration by Substitution & Separable Differential Equations.

Integration by Substitution Undoing the Chain Rule TS: Making Decisions After Reflection & Review.

2.1 Rates of Change and Limits. What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided.

Clicker Question 1 What is the volume of the solid formed when the curve y = 1 / x on the interval [1, 5] is revolved around the x-axis? – A.  ln(5) –

5 Logarithmic, Exponential, and Other Transcendental Functions

Integrating Exponential Functions TS: Making decisions after reflection and review.

Techniques of Integration Substitution Rule Integration by Parts Trigonometric Integrals Trigonometric Substitution Integration of Rational Functions by.

Techniques of Integration

Integration by Substitution Antidifferentiation of a Composite Function.

POLYNOMIALS - Evaluating When evaluating polynomials, we are simply substituting a value in for a variable wherever that variable appears in the expression.

Antiderivatives and Indefinite Integration. 1. Verify the statement by showing that the derivative of the right side equals the integrand of the left.

In this section, we introduce the idea of the indefinite integral. We also look at the process of variable substitution to find antiderivatives of more.

Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies.

U Substitution Method of Integration 5.5. The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives.

More with Rules for Differentiation Warm-Up: Find the derivative of f(x) = 3x 2 – 4x 4 +1.

5.6 Integration by Substitution Method (U-substitution) Thurs Dec 3 Do Now Find the derivative of.

5.6 Integration by Substitution Method (U-substitution) Fri Feb 5 Do Now Find the derivative of.

Test Review 1 § Know the power rule for integration. a. ∫ (x 4 + x + x ½ x – ½ + x – 2 ) dx = Remember you may differentiate to check your.

6.2 Antidifferentiation by Substitution Quick Review.

1 5.b – The Substitution Rule. 2 Example – Optional for Pattern Learners 1. Evaluate 3. Evaluate Use WolframAlpha.com to evaluate the following. 2. Evaluate.

7 TECHNIQUES OF INTEGRATION. As we have seen, integration is more challenging than differentiation. –In finding the derivative of a function, it is obvious.

5 Logarithmic, Exponential, and Other Transcendental Functions

Understanding Exponents

6 Integration Antiderivatives and the Rules of Integration

Lesson 4.5 Integration by Substitution

4.5 Integration by Substitution

Integration Techniques

PROGRAMME 16 INTEGRATION 1.

PROGRAMME 15 INTEGRATION 1.

Integration by Substitution

6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA

6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA

4.1 Antiderivatives and Indefinite Integration

The Fundamental Theorem of Calculus (FTC)

Copyright © Cengage Learning. All rights reserved.

Integral Rules; Integration by Substitution

Integration by Substitution (Section 4-5)

4.5 Integration by Substitution The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for.

integration is more challenging than differentiation.

Chapter 7 Integration.

PROGRAMME 17 INTEGRATION 2.

7.2 Antidifferentiation by Substitution

5 Logarithmic, Exponential, and Other Transcendental Functions

SUBSTITUTION At the end of this lesson you should :

Use long division to evaluate the integral. {image}

Techniques of Integration

Techniques of Integration

More Definite Integrals

Antidifferentiation by Substitution

Logarithmic, Exponential, and Other Transcendental Functions

Integration by Substitution

Presentation transcript:

Integrating Rational Functions TS: Making decisions after reflection and review

Objective To evaluate the integrals of exponential and rational functions.

Remember… What is the derivative of ln(u)? So what is the again?

Exponential Functions

Exponential Functions

Exponential Functions

Conclusion Integration by substitution is a technique for finding the antiderivative of a composite function. Experiment with different choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.