Explanation Given Information: The integral is ∫ x 3 − x 2 3 x 4 − 4 x 3 d x . Formula used: The constant multiple rule is: ∫ k ⋅ f ( x ) d x = k ∫ f ( x ) d x The special case of the power rule is: ∫ x − 1 d x = ln | x | + C For n = − 1 Calculation: Consider the integral, ∫ x 3 − x 2 3 x 4 − 4 x 3 d x Assume a function of u in terms of x from x 3 − x 2 3 x 4 − 4 x 3 , u = 3 x 4 − 4 x 3 Take the derivative d u d x of the function u = 3 x 4 − 4 x 3 , d u d x = 12 x 3 − 12 x 2 Solve for d x , d x = 1 12 x 3 − 12 x 2 d u = 1 12 ( x 3 − x 2 ) d u Now substitute u = 3 x 4 − 4 x 3 and d x = 1 12 ( x 3 − x 2 ) d u in the integral ∫ x 3 − x 2 3 x 4 − 4 x 3 d x , this gives, ∫ x 3 − x 2 3 x 4 − 4 x 3 d x = ∫ x 3 − x 2 u ⋅ 1 12 ( x 3 − x 2 ) d u = ∫ 1 12 u ( x 3 − x 2 ) ( x 3 − x 2 ) d u Simplify the integral by cancelling x 3 − x 2 by x 3 − x 2

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Author: Waner

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Chapter 6.2, Problem 32E

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To calculate: The solution of the integral ∫x3−x23x4−4x3dx .

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Chapter 6.2, Problem 31E

Chapter 6.2, Problem 33E

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The solution of the integral ∫ x 3 − x 2 3 x 4 − 4 x 3 d x .

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To calculate: The solution of the integral ∫x3−x23x4−4x3dx .

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Chapter 6 Solutions