Problem 54 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{1}^{e^{2}} \frac{3}{x} d x $$

Step-by-Step Solution

Answer

The value of the integral is 6.

Step 1: Understand the Problem

We need to evaluate the definite integral \( \int_{1}^{e^{2}} \frac{3}{x} \, dx \). This involves finding the integral of the function \( \frac{3}{x} \) from \( x = 1 \) to \( x = e^2 \).

Step 2: Identify the Indefinite Integral

The indefinite integral of \( \frac{3}{x} \) is \( 3 \ln |x| + C \), where \( C \) is the constant of integration.

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \) on an interval \([a, b]\), then: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] Here, \( F(x) = 3 \ln |x| \).

Step 4: Substitute the Limits of Integration

Substitute \( x = e^2 \) and \( x = 1 \) into \( F(x) = 3 \ln |x| \):\[ F(e^2) = 3 \ln |e^2| = 3 \ln e^2 \] \[ F(1) = 3 \ln |1| = 3 \ln 1 \] Recall, \( \ln 1 = 0 \).

Step 5: Simplify

Evaluate the expressions: - \( 3 \ln e^2 = 3 \times 2 = 6 \) because \( \ln e^2 = 2 \).- \( 3 \ln 1 = 0 \).Subtract these results: \[ \int_{1}^{e^2} \frac{3}{x} \, dx = 6 - 0 = 6 \]

Key Concepts

Fundamental Theorem of Calculus

Understanding the Fundamental Theorem of Calculus is crucial for evaluating definite integrals. This theorem bridges the concepts of a derivative and an integral, serving as a cornerstone in calculus. It consists of two main parts:

• The first part tells us that if we have a continuous function on an interval and an antiderivative for it, then the integral of this function over that interval can be found by evaluating the antiderivative at the bounds.


• Essentially, it states: if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).

In practice, this means once we find the antiderivative \(F(x)\), calculating the definite integral is as simple as plugging the upper and lower bounds into \(F(x)\) and subtracting.
This powerful method allows us to compute exact areas under curves efficiently using antiderivatives.

Indefinite Integral

An indefinite integral, often referred to as the antiderivative, is a central concept in integral calculus. When we calculate the indefinite integral of a function, we are essentially reversing the process of differentiation. Mathematically, this is represented as finding a function \(F(x)\) such that \(F'(x) = f(x)\).

• A common notation used is \(\int f(x) \, dx\), with the result being \(F(x) + C\).


• Here, \(C\) is the constant of integration, accounting for the fact that antiderivatives are not unique.

In the given problem, we recognize \(\frac{3}{x}\) as our function \(f(x)\), whose indefinite integral is \(3 \ln |x| + C\). This process transforms a derivative back into an original function plus a constant. Understanding indefinite integrals is essential because they form the foundation for solving definite integrals using the Fundamental Theorem of Calculus.

Natural Logarithm

The natural logarithm, expressed as \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. This function is the inverse of the exponential function \(e^x\).

• It simplifies many calculus-related expressions due to its unique properties.


• Notably, \(\ln(1)\) is always 0 because \(e^0 = 1\).


• Furthermore, \(\ln(e^x) = x\) due to the inverse relationship between logarithms and exponents.

In the problem at hand, the integral of \(\frac{3}{x}\) resulted in an expression involving \(3 \ln |x|\). Using properties of natural logarithms simplifies calculations, such as \(\ln(e^2) = 2\), making our definite integration results easier to evaluate. Mastery of natural logarithms is vital both for integral calculus and broader mathematical problem-solving.