Explanation Given Information: The provided definite integral is; ∫ 0 3 x ( 1 + 3 x ) 4 d x Formula used: The formula for the integral ∫ u ( a + b u ) 2 d u is: ∫ u ( a + b u ) 2 d u = 1 b 2 [ − 1 ( n − 2 ) ( a + b u ) n − 2 + a ( n − 1 ) ( a + b u ) n − 1 ] + C The General power differentiation Rule is : d d x [ u n ] = n u n − 1 d u d x Calculation: Consider u = x . Differentiate the considered function with respect to x using power rule of differentiation. d u = d x Consider the provided expression: ∫ 0 3 x ( 1 + 3 x ) 4 d x Here, a = 1 , b = 3 , n = 4 , u = x and d u = d x . Substitute u for x , a for 1 , b for 3 , n for 4 and d u for d x as;. ∫ 0 3 x ( 1 + 3 x ) 4 d x = ∫ 0 3 u ( a + b u ) n d x Use the above integration formula and solve the above integral as; ∫ 0 3 x ( 1 + 3 x ) 4 d x = [ 1 b 2 [ − 1 ( n − 2 ) ( a + b u ) n − 2 + a ( n − 1 ) ( a + b u ) n − 1 ] ] 0 3 Substitute x for u , 3 for b , 4 for n , and 1 for a

Calculus: An Applied Approach (Providence College: MTH 109)

9th Edition

ISBN: 9781285142616

Author: Ron Larson

Publisher: CENGAGE C

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Chapter 6.2, Problem 44E

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To calculate: The solution of definite integral ∫03x(1+3x)4dx.

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Chapter 6.2, Problem 43E

Chapter 6.2, Problem 45E

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Chapter 6 Solutions

Calculus: An Applied Approach (Providence College: MTH 109)

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The solution of definite integral ∫ 0 3 x ( 1 + 3 x ) 4 d x .

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To calculate: The solution of definite integral ∫03x(1+3x)4dx.

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Chapter 6 Solutions