Explanation Given: ∫ ln ( x 2 + 9 ) ⋅ d x Key Concept Applied: Integration by parts is a method for evaluating a difficult integral. When the integral is a product of functions, the integration by parts formula moves the products out the equation so the integral can be solved more easily. ∫ u ⋅ d v = u v − ∫ v ⋅ 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 ( g ( x ) ) where f ( x ) is the outside function and g ( 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. This integration rule follows from the chain rule for differentiation, because U-Substitution is nothing more than the reverse of the chain rule. Substitution Rule To integrate: ∫ f ( g ( x ) ) ⋅ g ' ( x ) ⋅ d x ↗ ↖ I n s i d e f u n c t i o n O u t s i d e F u n c t i o n ( D e r i v a t i v e o f I n s i d e ) The essence of the substitution rule is to take an integral in terms of x and transform or change it into terms of u by finding the outside function and inside function, where the outside function is the derivative of the inside function. Then we will make a suitable substitution that will simplify our integrand so that we can integrate, as illustrated in three easy steps below: ∫ f ( g ( x ) ) ⋅ g ' ( x ) ⋅ d x , I f u = g ( x ) ⇒ d u = g ' ( x ) ⋅ d x , T h e n d x = d u g ' ( x ) A f t e r U − S u b t i t u t i o n , I n t e g r a l w i l l b e c o m e ... ∫ f ( u ) ⋅ g ' ( x ) ⋅ d u g ' ( x ) C a n c e l l i n g g ' ( x ) , w e g e t ... ∫ f ( u ) ⋅ d u Formula used: Integration by parts ∫ u ⋅ d v = u v − ∫ v ⋅ d u Formulas of Common Derivative ( c ) ' = 0 , where c any constant ( c ⋅ x ) ' = c ⋅ d x Formulas of Common Integral ∫ 1 u ⋅ d u = ln | u | + c ∫ c f ( u ) ⋅ d u = c ⋅ ∫ f ( u ) ⋅ d u ∫ ( f ( u ) ± g ( u ) ) ⋅ d u = ∫ f ( u ) ⋅ d u ± ∫ g ( u ) ⋅ d u ∫ a ⋅ d x = a x Calculation: Description ∫ ln ( x 2 + 9 ) ⋅ d x Step 1: Rewrite the original integral in terms of u and v by substitution then apply the integration by parts formula ∫ u ⋅ d v = u v − ∫ v ⋅ d u ...

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8, Problem 57CRE

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Solution Summary: The author explains the integration by parts formula for evaluating a difficult integral. U-Substitution is the reverse of the chain rule for differentiation.

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To evaluate:

The given integral