Explanation Given: • ( a ) ∫ x ⋅ ( x 2 + 9 ) 4 ⋅ d x . • ( b ) ∫ x 2 ⋅ sin ( x 3 ) ⋅ d x . • ( c ) ∫ sin x ⋅ cos 2 x ⋅ d x Key Concepts Used: 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. 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 . U-Substitution is a technique we use when the integrand is a composite function. A composite function is a function inside another function of the form f ( u ( x ) ) where f ( x ) is the outside function and u ( x ) is the inside function. The first thing we will need to do is to recognize that we are being asked to integrate a product of a function and its derivative, and it takes the form of a composite function. This can be generalized for any product that is a composite function multiplied by the derivative of the inner function. Calculation: Description: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u Step 1 : Write the given integral ( a ) . Find out term which should be substituted by u . ∫ x ⋅ ( x 2 + 9 ) 4 ⋅ d x L e t u = x 2 + 9 d u d x = d d x ( x 2 + 9 ) = 2 x ⇒ d u 2 = x ⋅ d x Substitute these values in the given integral, ∫ x ⋅ ( x 2 + 9 ) 4 ⋅ d x = ∫ ( x 2 + 9 ) 4 ⋅ x ⋅ d x = ∫ u 4 ⋅ d u 2 = 1 2 ∫ u 4 ⋅ d u Thereby here our choice is u = x 2 + 9 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.7, Problem 2PQ

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An appropriate choice of $u$ for evaluating the given integrals by substitution

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Chapter 5.7, Problem 1PQ

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Answer the integral

Use U-Substitution to evaluate the definite integrals.

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

Chapter 5 Solutions

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