Explanation Given Information: The provided definite integral is ∫ 1 2 x 2 e 2 x d x . Formula used: The integration by parts ∫ u d v = u v − ∫ v d u . Where, u and v are function of x . ∫ e a x d x = e a x a + C Calculation: Consider the definite integral ∫ 1 2 x 2 e 2 x d x Here, d v = e 2 x d x and u = x 2 First find v , d v = e 2 x d x ∫ d v = ∫ e 2 x d x On further solving, v = e 2 x 2 …...…... (1) Now find d u , u = x 2 Differentiate both side with respect x , d u d x = d ( x 2 ) d x d u d x = 2 x And, d u = 2 x ⋅ d x …...…... (2) Apply integration by parts formula and substitute equation (1) and (2) in ∫ u d v = u v − ∫ v d u , ∫ x 2 e 2 x d x = 1 7 x 2 e 2 x − 2 2 ∫ x e 2 x d x Use integration by parts on the second integral ∫ x e 2 x d x Here, d v = e 2 x d x and u = x First find v , d v = e 2 x d x ∫ d v = ∫ e 2 x d x On further solving, v = e 2 x 2 …...…... (3) Now find d u , u = x Differentiate both side with respect x , d u d x = d ( x ) d x d u d x = 1 And, d u = 1 ⋅ d x …

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.1, Problem 45E

To determine

To calculate: The value of definite integral ∫12x2e2xdx.

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Chapter 6 Solutions

Calculus: An Applied Approach (MindTap Course List)

Ch. 6.1 - Prob. 1CP Ch. 6.1 - Prob. 2CP Ch. 6.1 - Prob. 3CP Ch. 6.1 - Prob. 4CP Ch. 6.1 - Prob. 5CP Ch. 6.1 - Prob. 6CP Ch. 6.1 - Prob. 7CP Ch. 6.1 - Prob. 1SWU Ch. 6.1 - Prob. 2SWU Ch. 6.1 - Prob. 3SWU

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The value of definite integral ∫ 1 2 x 2 e 2 x d x .

Solution Summary: The author explains the formula used to calculate the value of the definite integral, which is underset_66.37.

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To calculate: The value of definite integral ∫12x2e2xdx.

Chapter 6 Solutions