   Chapter 6, Problem 10RE ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Evaluating a Definite integral In Exercises 9–12, use integration by parts to evaluate the definite integral. ∫ 0 4 ln ( 1 + 3 x ) d x

To determine

To calculate: The definite integral of 04ln(1+3x)dx by using the method of integration by parts.

Explanation

Given Information:

The provided integral is 04ln(1+3x)dx.

Formula used:

The method of integration by parts:

If v and u are two differentiable function of x. Then,

udv=uvvdu

Steps to solve the integral problems:

Step1: At first find the most complicated portion of the integrand and try to letter it as dv so that it can fit a fundamental integration rule. Then, the remaining factor or factors of the integrand will be u.

Step2: First find the factor whose derivative is simple and consider it as u and then the remaining factor or factors of the integrand will be dv and dv should always include the term dx of the original integrand.

Calculation:

Recall the provided integral.

04ln(1+3x)dx

In the above integrand, the simplest portion of the integrand is ln(1+3x). So, consider, u=ln(1+3x) and the remaining factors as dv=dx.

Therefore,

du=11+3x(3)dx=31+3xdx

And,

dv=dx

Integrate the above expression for v.

dv=dxv=x

Again, apply the integration by parts.

udv=uvvdu

Substitute ln(1+3x) for u, x for v, dx for dv and 31+3xdx for du

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