Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
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Chapter 5.7, Problem 21E
To determine

To calculate: The value of given integral

Expert Solution & Answer
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Answer to Problem 21E

The value of the given integral is 12ln|2x+1|+2ln|x−1|+C

Explanation of Solution

Given information:

The integral is ∫5x+1(2x+1)(x−1)dx

Formula used:

  ∫1xdx=ln|x|+C

Calculation:

The denominator is factored in the given integral so do the partial function expansion of the integrand

  5x+1(2x+1)(x−1)=A2x+1+Bx−1

Here, A and B are constants. To find the values A and B multiply both sides of the equation by (2x+1)(x−1)

  5x+1=A(x−1)+B(2x+1)5x+1=Ax−A+2Bx+B5x+1=x(A+2B)+(−A+B)

The coefficients of x must be equal and the constant terms are also equal. So

  A+2B=5 and −A+B=1⇒B=1+A

Solve this system of linear equation for A and B , by use of simple substitution

  5=A+2(1+A) =3A+2

So, A=1 and B=2

Rewrite the integral using the values found for A and B in the previous step

  ∫12x+1dx+∫2x−1dx

Each of the resultant partial fraction is easy to integrate, use the substitution u=2x+1 and v=x−1 respectively. Note that du2=dx and du=dv for this substitution. So,

  12∫1udu+2∫1vdv=12ln|u|+2ln|v|+C

Integrate the simplified integrals. Here, C is the arbitrary constant. Substitute the values of u and v back into the equation to get the final answer

  ∫5x+1(2x+1)(x−1)dx=12ln|u|+2ln|v|+C=12ln|2x+1|+2ln|x−1|+C

Therefore, the value of the given integral is 12ln|2x+1|+2ln|x−1|+C

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Chapter 5 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 5.1 - Prob. 1ECh. 5.1 - (a) Use six rectangles to find estimates of each...Ch. 5.1 - (a) Estimate the area under the graph of f(x) =...Ch. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - The speed of a runner increased steadily during...Ch. 5.1 - Prob. 12E
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