Problem 18 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(\frac{x}{x-4}\right) d x\)

Step-by-Step Solution

Answer

x + 4 ln|x-4| + C

Step 1: Simplify the Integrand

Rewrite the integrand \(\frac{x}{x-4}\) using algebraic manipulation. Split the fraction: \(\frac{x}{x-4} = 1 + \frac{4}{x-4}\).

Step 2: Break the Integral into Two Parts

Express the integral as the sum of two integrals: \(\begin{aligned} \ \int\frac{x}{x-4} \ d x &= \int 1 \ d x + \int\frac{4}{x-4} \ d x \ \end{aligned}\).

Step 3: Integrate Each Part Separately

Integrate \(\begin{aligned} \ \int 1 \ d x &= x + C_1 \ \int\frac{4}{x-4} \ d x &= 4 \int\frac{1}{x-4} \ d x = 4 \ln|x-4| + C_2 \ \end{aligned}\).

Step 4: Combine the Results

Combine the results from the previous step: \(\begin{aligned} \ \int\frac{x}{x-4} \ d x &= x + 4 \ln|x-4| + C \ \end{aligned}\). Here, \(C = C_1 + C_2\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation helps simplify complex expressions, making them easier to integrate. In the given exercise, we start with the integrand \(\frac{x}{x-4}\). To simplify this, we use the strategy of splitting the fraction.

First, recognize that \(\frac{x}{x-4}\) can be rewritten. Break it apart using basic algebraic principles: \(\frac{x}{x-4} = 1 + \frac{4}{x-4}\).

This step is crucial because it translates a more complicated fraction into simpler parts that are easier to integrate.

Integral Splitting

Integral splitting divides a complex integral into simpler, more manageable pieces. After algebraically manipulating the integrand, we express it as a sum of separate terms.

For the integrand \(\frac{x}{x-4} = 1 + \frac{4}{x-4}\), we split the integral:

\[ \int \frac{x}{x-4} \ dx = \int 1 \ dx + \int \frac{4}{x-4} \ d x \]

By breaking it down, it becomes easier to tackle each part individually.

Logarithmic Integration

Logarithmic integration comes into play when dealing with integrals of the form \(\frac{1}{x+a}\). Recall that the integral of \(\frac{1}{x}\) is \(\text{ln}|x|\).

For our integral, you integrate the two parts separately:

\[ \int 1 \ dx = x + C_1 \]
\[ \int \frac{4}{x-4} \ dx = 4 \int \frac{1}{x-4} \ dx = 4 \text{ln}|x-4| + C_2 \]

Logarithmic rules make the second integral straightforward.

Constants of Integration

Any indefinite integral includes a constant of integration. These constants arise because antiderivatives are not unique; they can vary by a constant.

During integration, two constants appear: \C_1\ and \C_2\. When combining results, we consolidate them into a single constant, \C\.

Summing things up:

\[ \int \frac{x}{x-4} \ dx = x + 4 \text{ln}|x-4| + C \]

Here, \C = C_1 + C_2\. This final constant accounts for all arbitrary constants from individual integrals.