Problem 37

Question

$$ \text { In Problems } 37-44, \text { evaluate each definite integral. } $$ $$ \int_{3}^{5} \frac{x-1}{x} d x $$

Step-by-Step Solution

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Answer

The value of the integral is $ 2 - \ln\left(\frac{5}{3}\right) $.

1Step 1: Rewrite the Integrand

Rewrite the integrand $ \frac{x-1}{x} $ as a difference of two fractions: $ \frac{x}{x} - \frac{1}{x} = 1 - \frac{1}{x} $. This simplification makes it easier to integrate.

2Step 2: Set Up the Integral

Set up the integral of the simplified expression: $ \int_{3}^{5} \left(1 - \frac{1}{x}\right) \, dx $. This will make it easier to find the antiderivative.

3Step 3: Integrate the Expression

Integrate each part of the expression separately. The integral of $ 1 $ with respect to $ x $ is $ x $, and the integral of $ -\frac{1}{x} $ with respect to $ x $ is $ -\ln|x| $. Thus, the antiderivative of the expression is $ x - \ln|x| $.

4Step 4: Evaluate the Definite Integral

Apply the limits of integration from 3 to 5 to the antiderivative. Evaluate $ \left[ x - \ln|x| \right]_{3}^{5} = \left( 5 - \ln(5) \right) - \left( 3 - \ln(3) \right) $.

5Step 5: Simplify the Result

Simplify the expression to find the final result: $ (5 - 3) - (\ln(5) - \ln(3)) = 2 - \ln\left(\frac{5}{3}\right) $.

Key Concepts

Understanding AntiderivativesIntegrand SimplificationImportance of Limits of Integration

Understanding Antiderivatives

An antiderivative, also known as an indefinite integral, is a function that reverses what the derivative does. Finding an antiderivative means identifying a function whose derivative gives you the original function inside the integral. In our step-by-step solution, the task was to integrate the expression $ \int \left(1 - \frac{1}{x}\right) \, dx $.

Here's how it works:

The antiderivative of the constant $ 1 $ with respect to $ x $ is $ x $, because taking the derivative of $ x $ gives you back $ 1 $.
The antiderivative of $ -\frac{1}{x} $ is $ -\ln|x| $, since the derivative of $ \ln|x| $ is $ \frac{1}{x} $.

Putting it together, the antiderivative of our original integrand is $ x - \ln|x| $. This function allows us to easily evaluate definite integrals.

Integrand Simplification

Simplifying the integrand is often a crucial first step in making an integral easier to solve. In this exercise, we started with $ \frac{x-1}{x} $. To simplify, we broke it down into two separate fractions: $ \frac{x}{x} $ and $ -\frac{1}{x} $.

This results in the expression $ 1 - \frac{1}{x} $, which is much simpler to work with when integrating.

$ \frac{x}{x} $ simplifies to $ 1 $, simply stating that any number divided by itself equals one (unless it's zero).
The term $ \-\frac{1}{x} $ remains unchanged but is now clearly defined as a distinct part of the function to integrate.

Simplifying the integrand allows for straightforward integration and sets up the problem in a simple form where basic rules of integration can be easily applied.

Importance of Limits of Integration

Limits of integration define the start and end points of a definite integral, giving it specific numerical bounds. Unlike indefinite integrals, definite integrals result in a specific value rather than a general formula plus a constant. Here, the limits are from 3 to 5.

To evaluate the definite integral, we first find the antiderivative of the integrand. Then we apply the Fundamental Theorem of Calculus by evaluating the antiderivative at these bounds:

Substitute 5 into the antiderivative $ x - \ln|x| $ to get $ 5 - \ln(5) $.
Substitute 3 into the same expression to get $ 3 - \ln(3) $.

Subtract the second result from the first to find the value of the definite integral, which is $ 2 - \ln\left(\frac{5}{3}\right) $.

The limits of integration provide a precise answer to how much "area" is under the curve of the function between $ x = 3 $ and $ x = 5 $. Often, the definite integral can represent physical quantities, such as total distance, area, or even probability, making limits of integration key to determining specific numerical answers.

Problem 37

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