Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x-1}{4 x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function \( \frac{x-1}{4x} \) is \( \frac{1}{4}x - \frac{1}{4}\ln |x| + C \)

Step 1: Simplify the integrand

First, simplify the integrand by applying polynomial division in the fraction \( \frac{x-1}{4x} \). This results in \( \frac{1}{4} - \frac{1}{4x} \).

Step 2: Separate the integrals

Next, express the integrand as a sum of simpler fractions, which can be integrated separately: \[ \int\frac{1}{4}dx - \int\frac{1}{4x}dx \].

Step 3: Apply basic integration rules

The given expressions are now basic formulas in integration. The integral of a constant \(k\) is \(kx\), and the integral of \( \frac{1}{x} \) is \( \ln |x| \), so apply these rules to each term. This gives us: \[ \frac{1}{4}x - \frac{1}{4}\ln |x| + C .\] Note that the constant in multiplication can be factored out of the integrals, which adds simplicity to the computations. C here denotes the constant of integration.

Key Concepts

Polynomial Division

When you encounter a rational expression like \( \frac{x-1}{4x} \), sometimes simplifying it with polynomial division can make integration easier. In polynomial division, the numerator (dividend) is divided by the denominator (divisor) to give a quotient and sometimes a remainder. This process is somewhat similar to long division but used for polynomials.

In our case, dividing \( x - 1 \) by \( 4x \) simplifies to \( \frac{1}{4} - \frac{1}{4x} \). Here’s how it works:

• Firstly, see that \( 4x \) goes into \( x \) with a coefficient of \( \frac{1}{4} \).
• Subtract \( \frac{1}{4} \times 4x \) from \( x - 1 \), which simplifies to \( -1 \).
• The remaining part \( -1 \) must now be divided by the original \( 4x \), resulting in \( -\frac{1}{4x} \).

This simplification converts our complex rational expression into something much more manageable for integration. By doing so, we can then deal with simpler terms separately.

Basic Integration Rules

Integration can seem daunting, but thankfully, certain basic rules help us. Let's break down the rules applied in this exercise.

When faced with \( \int \frac{1}{4} \, dx \) and \( \int \frac{1}{4x} \, dx \), we use these fundamental integration rules:

• The integral of a constant \( k \) with respect to \( x \) is really straightforward: it is \( kx \). So, \( \int \frac{1}{4} \, dx \) becomes \( \frac{1}{4}x \).
• The integral of \( \frac{1}{x} \) is a special case that leads to a natural logarithm, specifically \( \ln |x| \). Therefore, \( \int \frac{1}{4x} \, dx \) transforms to \( \frac{1}{4} \ln |x| \).

Each of these expressions is simplified by factoring out the constant \( \frac{1}{4} \) from the integral, joining them to form a final solution that incorporates both expressions.

Constant of Integration

Whenever we calculate indefinite integrals, you'll notice a \( C \) added to the solution—the constant of integration. Wondering why it's there? Let's clarify!

The constant of integration is vital because every time you integrate a function, you’re essentially finding a family of functions. All these possible functions differ by a constant. Since differentiation of a constant is zero, it does not appear when you find the derivative, and hence any constant could have been part of the original function before integration.

• For example, if the derivative of a function \( F(x) \) gives \( f(x) \), then integrating \( f(x) \) should give you \( F(x) + C \).
• This ensures that constant unknowns are accounted for, providing the flexibility needed when working with indefinite integrals.

In our exercise, after using polynomial division and basic integration rules, remember to add \( + C \) to the final expression \( \frac{1}{4}x - \frac{1}{4}\ln |x| + C \). This guarantees that the solution encompasses every potential initial condition.