Explanation Given: • Algebraic manipulation • Substitution (Specify u a n d d u ) • Integration by parts (Specify u a n d d v ) • Integral: ∫ x 3 4 − x 2 ⋅ d x Key Concepts Used: We know very well that integration is more challenging than differentiation. In differentiation, we know to find the derivative of a function by using the correct differentiation formula. But in integration, we should not confirm which technique we should use to integrate a given function. We can use substitution, integration by parts, and partial fractions. Definite Integral: The definite integral of f ( x ) over [ a , b ] is denoted by the integral sign ∫ f ( x ) ⋅ d x a b . It is often called the integral of f ( x ) over [ a , b ] . The process of computing integrals is called Integration. The function f ( x ) is called the integrand. The endpoints a a n d b of [ a , b ] are called the Limits of Integration. If f ( x ) is continuous on [ a , b ] , or if f ( x ) is continuous with at most finitely many jump discontinuous, then it is possible to integrate f ( x ) over [ a , b ] . First we should thorough with the basic integration formulas. Then we might try the four-step strategy. 1. Simplify the integrand if possible: Sometimes the use of algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious. 2. Look for an obvious substitution: Try to find some function u = g ( x ) in the integrand whose differential d u = g ' ( x ) ⋅ d x also occurs, apart from a constant factor. We can use substitution instead of method of partial fractions. 3. Classify the Integrand according to its form: If steps 1 a n d 2 have not led to the solution, then we have to see the form of the integrand. (a) Trigonometric functions: If f ( x ) is a product or powers of sin x , cos x , tan x , csc x , sec x o r cot x , then we find the integral by using substitution process. (b) Rational functions: If f ( x ) is a rational function, we use procedure involving partial fractions (algebraic manipulation) (c) Integration by parts: If f ( x ) is a product of power of x (or) a polynomial, and a transcendental function ( such as a trigonometric, exponential, inverse trigonometric, or logarithmic function) then we try integration by parts, choosing u a n d d u and in final replace the dummy variable u by original variable. (d) Radicals: Particular kinds of substitutions are recommended when certain radicals appear. (i) If ± x 2 ± a 2 occurs, we use a trigonometric substitution. (ii) If ± x 2 ± a 2 n occurs, we use the rationalizing substitution u = a x + b n . 4. Try again: If the first three steps have not produced the answer, then we have to remember that there are basically only two methods of integration: substitution and integration by parts. Substitution Method: When the integrand has the form f ( u ( x ) ) ⋅ u ' ( x ) , we can use the equation d u = u ' ( x ) ⋅ d x to rewrite the entire integral (including the d x term) in terms of u and it’s differential d u as follows: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u Algebraic Manipulation: In algebraic manipulation there are four techniques which we commonly use...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.1, Problem 75E

To determine

To Describe:

A good method to evaluate given integral (no need to evaluate)

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Chapter 8.1, Problem 74E

Chapter 8.1, Problem 76E

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indicate a good method for evaluating the integral (but do not evaluate). Your choices are algebraic manipulation, substitution (specify u and du), and Integration by Parts (specify u and dv). If it appears that the techniques you have learned thus far are not sufficient, state this.

1) Can you calculate this integral using the substitution method? 2) consider the integral 3) calculate xe^x dx by calculating the integral that appears in your answer to question 2

Solve for the value of the double integral below and decipher if Fubini's theorem applies to it.

Chapter 8 Solutions

CALCULUS (CLOTH)

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - Prob. 55E Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - 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A good method to evaluate given integral (no need to evaluate)

Solution Summary: The author explains how to evaluate a given integral using substitution, integration by parts, and partial tions.

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To Describe: A good method to evaluate given integral (no need to evaluate)

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A good method to evaluate given integral (no need to evaluate)