Problem 9 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Rewrite the integral with the decomposed function

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

Step 4: 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.