Explanation Given Information: The expression is ∫ 0 ln 3 x 2 e 2 x d x . Formula used: The formula for the integration by part is as follows: ∫ u d v = u v − ∫ v d u Calculation: Consider the following expression: ∫ x 2 e 2 x d x Assume u = x 2 . Determine d u by taking its derivative: d u = 2 x d x Assume d v = e 2 x d x . Determine v by taking its 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 x 2 for u , e 2 x d x for d v , 1 2 e 2 x for v and 2 x d x for d u . ∫ x 2 e 2 x d x = x 2 ⋅ 1 2 e 2 x − ∫ 1 2 e x ⋅ 2 x d x = 1 2 x 2 e 2 x − ∫ x e 2 x d x …… (1) Use the integration by parts for second time. Now, consider the expression: ∫ x e 2 x d x Assume U = x . Determine d U by taking its derivative: d U = d x Assume d V = e 2 x d x . Determine V by taking its anti-derivative. d V = ∫ e x d x V = 1 2 e 2 x Recall the integration by part formula: ∫ U d V = U V − ∫ V d U Substitute x for U , e 2 x d x for d V , 1 2 e 2 x for V and d x for d U . ∫ x e 2 x d x = x ⋅ 1 2 e 2 x − ∫ 1 2 e 2 x d x = 1 2 x e 2 x − 1 2 ⋅ 1 2 e 2 x + K = 1 2 x e 2 x − 1 4 e 2 x + K Substitute 2 x e x − 2 e x + K for ∫ 2 x e x d x in equation (1)

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 4.6, Problem 38E

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To calculate: The value of the expression ∫0ln3x2e2xdx by using integration by parts.

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Chapter 4.6, Problem 37E

Chapter 4.6, Problem 39E

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The value of the expression ∫ 0 ln 3 x 2 e 2 x d x by using integration by parts.

Solution Summary: The author explains how to calculate the value of the expression displaystyle 'underset' by using integration by parts.