Explanation Given: The expression is ∫ 4 x 2 e 2 x d x . Formula used: Write the integration by part formula: ∫ u d v = u v − ∫ v d u Calculation: Consider the following expression: ∫ 4 x 2 e 2 x d x Assume u = 4 x 2 . Determine d u by take a derivative: d u = 8 x d x Assume d v = e 2 x d x . Determine v by taking an anti-derivative. ∫ d v = ∫ e 2 x d x v = 1 2 e 2 x Recall the integration by part formula: ∫ u d v = u v − ∫ v d u Substitute 4 x 2 for u , e 2 x d x for d v , 1 2 e 2 x for v and 8 x d x for d u . ∫ 4 x 2 e 2 x d x = 4 x 2 ( 1 2 e 2 x ) − ∫ 1 2 e 2 x 8 x d x = 2 x 2 e 2 x − ∫ 4 x e 2 x d x …… (1) Now, consider the expression: ∫ 4 x e 2 x d x Assume U = 4 x . Determine d U by take a derivative: d U = 4 d x Assume d V = e 2 x d x . Determine V by taking an anti-derivative

MyLab Math with Pearson eText -- Standalone Access Card -- for Mathematics with Applications in the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134856551

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 13.3, Problem 3CP

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To calculate: The value of the expression ∫4x2e2xdx by using the integration by parts twice.

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Chapter 13.3, Problem 2CP

Chapter 13.3, Problem 4CP

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Trigonometric Substitution There are certain integrals that contain no trigonometry that can be solved with the aid of trigonometry. Consider integrands involving va? -x, a? +x, or vx - a (where a is constant). These can be evaluated by making the following substitutions. Va? + x x= asin 6 x= atan 0 X= asec 0 dx = acos ede dx = a sec*o de dx = asec Otan ede For each integral, figure out which trigonometric substitution should be used. The goal is to rewrite the integrand so that there is no radical. After integrating, draw and label a right triangle (as shown below). This will allow you to rewrite your integrated expression back in terms of x. For this last part of the problem, you may need one or more trigonometric identities. For example, the triangle that corresponds with exercise 1 looks like this. If we let x = 2sin 0, it follows that sin 6 = x/2. By expressing the side length opposite the given angle as x and the hypotenuse as 2, the side length adjacent the given angle works out…

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Chapter 13 Solutions

MyLab Math with Pearson eText -- Standalone Access Card -- for Mathematics with Applications in the Management, Natural, and Social Sciences (12th Edition)

Ch. 13.1 - Checkpoint 1 Find an antiderivative for each of...Ch. 13.1 - Checkpoint 2 Find each of the...Ch. 13.1 - Prob. 3CP Ch. 13.1 - Prob. 4CP Ch. 13.1 - Prob. 5CP Ch. 13.1 - Prob. 6CP Ch. 13.1 - Prob. 7CP Ch. 13.1 - Checkpoint 8 The marginal cost at a level of...Ch. 13.1 - 1. What must be true of F(x) and G(x) if both are...Ch. 13.1 - 2. How is the antiderivative of a function related...

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The value of the expression ∫ 4 x 2 e 2 x d x by using the integration by parts twice.

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To calculate: The value of the expression ∫4x2e2xdx by using the integration by parts twice.

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Chapter 13 Solutions