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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using Integration Tables In Exercises 37– 44, use the integration table in Appendix C to evaluate the definite integral. See Example 4.

0 3 x ( 1 + 3 x ) 4 d x

To determine

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

Explanation

Given Information:

The provided definite integral is;

03x(1+3x)4dx

Formula used:

The formula for the integral u(a+bu)2du is:

u(a+bu)2du=1b2[1(n2)(a+bu)n2+a(n1)(a+bu)n1]+C

The General power differentiation Rule is :

ddx[un]=nun1dudx

Calculation:

Consider u=x.

Differentiate the considered function with respect to x using power rule of differentiation.

du=dx

Consider the provided expression:

03x(1+3x)4dx

Here, a=1, b=3, n=4, u=x and du=dx.

Substitute u for x, a for 1, b for 3, n for 4 and du for dx as;.

03x(1+3x)4dx=03u(a+bu)ndx

Use the above integration formula and solve the above integral as;

03x(1+3x)4dx=[1b2[1(n2)(a+bu)n2+a(n1)(a+bu)n1]]03

Substitute x for u, 3 for b, 4 for n, and 1 for a

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