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Chapter 11: Problem 30

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

Short Answer

Expert verified
The short answer to the question "Evaluate the definite integral: \(\int_{1}^{3} \frac{2}{x} d x\)" would be: \[\int_{1}^{3} \frac{2}{x} d x = 2\ln 3\]

Step by step solution

01

Find the antiderivative of the given function

Let's first find the antiderivative of the given function, $$\frac{2}{x}$$. Recall the integration rule for functions of the form \(\int x^n dx\) with \(n \neq -1\) is: \[\int x^n d x = \frac{x^{n+1}}{n+1} + C\] However, in this case, \(n = -1\). For functions with \(n=-1\), the antiderivative is simply the natural logarithm: \[\int x^{-1} d x = \int \frac{1}{x} d x = \ln |x| + C\] Now, we will apply this rule to the function $$\frac{2}{x}$$ which is equal to $$2 \cdot \frac{1}{x}$$. \[\int \frac{2}{x} d x = 2 \left(\int \frac{1}{x} d x\right) = 2 \ln |x| + C\]
02

Use the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, when solving a definite integral, we need to find the antiderivative (as we did in Step 1) and then subtract the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit. In this case, the definite integral is given by: \[\int_{1}^{3} \frac{2}{x} d x = 2\ln |3| - 2\ln |1|\]
03

Evaluate the definite integral

Now, let's evaluate the expression we found in Step 2: \[2\ln |3| - 2\ln |1| = 2\ln 3 - 2\ln 1\] Since \(\ln 1 = 0\), the expression simplifies to: \[2\ln 3 - 2(0) = 2\ln 3\] Thus, the definite integral of the given function is: \[\int_{1}^{3} \frac{2}{x} d x = 2\ln 3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
An antiderivative, sometimes referred to as an indefinite integral, is essentially a reverse operation to differentiation. For a given function, an antiderivative is another function whose derivative is the given function. This means if you have a function, say, f(x), and you compute its derivative to get f'(x), then an antiderivative of f'(x) would be f(x) plus an arbitrary constant, typically denoted as C. The constant C represents all possible vertical shifts of the antiderivative on a graph.

In the step by step solution, it is clearly explained that the antiderivative of the function 2/x, which can be expressed as 2*x^-1, is 2 ln |x| + C. This is because the function 1/x is a special case where the exponent of x is exactly -1. The usual rule for integration of x^n does not apply, and instead, the natural logarithm function comes into play, signifying the unique nature of inverse power functions in calculus.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, serving as one of the core pillars of calculus. It has two parts, with the first part essentially stating that given an integrable function f(x), there exists an antiderivative F(x), such that F'(x) = f(x). The second part of the theorem is particularly useful for computing definite integrals: it states that if F(x) is an antiderivative of f(x) on the interval [a, b], then the definite integral of f(x) from a to b is equal to F(b) - F(a).

Using this theorem, when we evaluate the definite integral from 1 to 3 for the function 2/x, we simply take the difference of the antiderivative at 3 and 1. In the given solution, this evaluation step is executed flawlessly, providing students with a clear method to solve similar problems.
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. This special logarithm function is the inverse operation of taking the power of e, symbolized as e^x. The natural logarithm is particularly important in calculus because it is the integral of 1/x when x is greater than 0. Additionally, the derivative of ln(x) is 1/x, confirming its vital role in differentiation and integration.

Natural logarithms are also pivotal because of their natural appearance in growth/decay processes and many areas of physics, economics, and engineering. In the example exercise, the function 2/x is integrated to result in 2 ln(3), representing the natural logarithm's presence in evaluating the definite integral.

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