(a) Explanation Given: The granted indefinite integral is ∫ x 3 4 + x 2 d x and take d v = x 4 + x 2 d x Formula used: The integration of: ∫ t n d t = t n + 1 n + 1 The integration by parts: ∫ u d v = u v − ∫ v d u The differentiation of d d x x n = n x n − 1 Calculation: To integrate ∫ x 3 4 + x 2 d x by parts, choose u = x 2 , so that d v = x 4 + x 2 d x Now, integrate on both sides as follows: d v = x 4 + x 2 d x ∫ d v = ∫ x 4 + x 2 d x …...…... (1) Put 4 + x 2 = t and differentiate with respect to x as follows: d d x ( 4 + x 2 ) = d t d x 2 x = d t d x 2 x d x = d t ( x ) d x = 1 2 d t …...…... (2) Substitute (2) in (1) as follows: ∫ d v = 1 2 ∫ d t t ∫ d v = 1 2 ∫ t − 1 2 d t ∫ d v = 1 2 t − 1 2 + 1 ( − 1 2 + 1 ) + C ∫ d v = t + C ∫ d v = 4 + x 2 + C …...…... (3) Further, evaluating as follows: ∫ x 3 4 + x 2 d x = ∫ ( x 2 ) ( x 4 + x 2 d x ) = ∫ ( x 2 ) ( d v ) … (b) To determine To calculate: The expression for the indefinite integral ∫ x 3 4 + x 2 d x by substitution method.

Bundle: Calculus, 10th + WebAssign Printed Access Card for Larson/Edwards' Calculus, 10th Edition, Multi-Term

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ISBN: 9781285338231

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.2, Problem 63E

(a)

To determine

To calculate: The expression for the indefinite integral ∫x34+x2dx by integration by parts.

(b)

To determine

To calculate: The expression for the indefinite integral ∫x34+x2dx by substitution method.

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Chapter 8.2, Problem 58E

Chapter 8.2, Problem 64E

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The expression for the indefinite integral ∫ x 3 4 + x 2 d x by integration by parts.

Solution Summary: The author explains how to calculate the expression for the indefinite integral 'displaystyle int' by integration by parts.