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Chapter 12: Problem 9

Use the table of integrals in this section to find the indefinite integral. $$ \int \frac{1}{x(1+x)} d x $$

Short Answer

Expert verified
The indefinite integral \(\int \frac{1}{x(1+x)} dx = ln|x| - ln|1+x| + C\).

Step by step solution

01

Perform partial fraction decomposition

First, it's necessary to rewrite the integrand as the sum of simpler fractions through the process of partial fraction decomposition. As the degree of the numerator is less than the degree of the denominator, the integrand is already proper. So, let's express the integrand \(\frac{1}{x(1+x)}\) as \(\frac{A}{x} + \frac{B}{1+x}\). Our task is finding the correct values of A and B.
02

Determine coefficients (A, B)

In order to find A and B, we get rid from denumerators by multiplying the whole equation by the common denominator \(x(1+x)\) which results in \(1 = A(1+x) + Bx\). For \(x=0\), the equation reduces to \(1=A\), and for \(x=-1\), it reduces to \(1=-B\). Therefore, A equals 1 and B equals -1.
03

Rewrite the integral with the decomposed function

Now the integral can be rewritten as the sum of two simpler integrals: \[\int \frac{1}{x(1+x)} dx = \int \frac{1}{x} dx - \int \frac{1}{1+x} dx\]
04

Evaluate the integrals

The integral of \(\frac{1}{x}\) is \(ln|x|\) and the integral of \(\frac{1}{1+x}\) is \(ln|1+x|\). Therefore, the solution is: \[\int \frac{1}{x(1+x)} dx = ln|x| - ln|1+x| + C\], where C is the constant of integration.

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Most popular questions from this chapter

Population Growth \(\ln\) Exercises 57 and 58, use a graphing utility to graph the growth function. Use the table of integrals to find the average value of the growth function over the interval, where \(N\) is the size of a population and \(t\) is the time in days. $$ N=\frac{375}{1+e^{4.20-0.25 t}}, \quad[21,28] $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{1}\left(\frac{x^{2}}{2}+1\right) d x, n=4 $$

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.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=8 $$

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$
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