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Chapter 12: Problem 1

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{x}{(2+3 x)^{2}} d x, \text { Formula } 4 $$

Short Answer

Expert verified
The indefinite integral \(\int \frac{x}{(2+3x)^2} dx\) evaluates to \(-\frac{x}{3(2+3x)}+\frac{1}{3}\ln|2+3x| + C\).

Step by step solution

01

Identify the Integral Formula

The first step is to recognize the type of the integrand and correspond it with the appropriate formula. As follows: \[\int u dv = uv - \int v du\] This is referred to as 'Formula 4' or the formula for integration by parts.
02

Identification of Coefficients

We identify the value of \(u, du, v\) and \(dv\). In our integral \(\int \frac{x}{(2+3x)^2} dx\), we can take \(u=x\), so \(dv=\frac{1}{(2+3x)^2}dx\). Therefore, \(du=dx\). For \(v\), we evaluate the integral of \(dv\), that is, \(\int \frac{1}{(2+3x)^2}dx\), which turns out to be \(-\frac{1}{3(2+3x)}\). Note that a substitution method is used to find the integral of \(dv\), by setting \(t=(2+3x)\), and then finding the integral of \(\frac{1}{t^2}\).
03

Use the Formula to Evaluate the Integral

Using the formula, the integral can be evaluated as: \[\int \frac{x}{(2+3x)^2} dx = uv - \int v du = x \left(-\frac{1}{3(2+3x)}\right) - \int -\frac{1}{3(2+3x)} dx\]Simplify it to: \[-\frac{x}{3(2+3x)}+\frac{1}{3}\int \frac{1}{2+3x} dx\]The integral on the right side can be solved using straightforward integration and the result turns out to be \(\frac{1}{3}\ln|2+3x|\).
04

Combine the Results

The result from the previous step is now combined to give us the final answer:\[-\frac{x}{3(2+3x)}+\frac{1}{3}\ln|2+3x| + C\]where \(C\) is the constant of integration.

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Most popular questions from this chapter

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=4 $$

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{2}^{5} 10 x e^{-x} d x $$

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