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...

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.1, Problem 69E

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

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Chapter 8.1, Problem 68E

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Evaluate +1+ x2 dx. (1+ x²)² Hint: Write this as two "easy" integrals. You do not need u-substitution or integration by-parts. 1

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Calculus - Standalone book

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

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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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Chapter 8 Solutions

To Describe:

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