Question

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with \(C=0\) over the given interval \([a, b] .\) Approximate a value of \(C\), if possible, such that adding \(C\) to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{1}{x \sqrt{x^{2}-4}} d x\) over \([2,6]\)

Short answer

Set the graph with antiderivative \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | \) and \(C=0\) to visualize the change over [2,6].

Step-by-step solution

Identify the Function to Integrate

We need to find the antiderivative of the function \(f(x) = \frac{1}{x \sqrt{x^{2}-4}}\) and graph it over the interval \([2, 6] \).

Perform the Integration

The integral of \(f(x) = \frac{1}{x \sqrt{x^{2}-4}}\) is a standard form that can be recognized. The antiderivative is \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | + C \).

Set up to Plot the Antiderivative

Using a calculator or graphing tool, plot the function \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | \) using \(C=0\) over the interval \([2, 6]\).

Evaluate the Definite Integral

Compute the definite integral \( \int_{2}^{x} \frac{1}{t \sqrt{t^{2}-4}} dt \). For the approximate evaluation over the interval \([2, 6]\), consider \( F(6) = \frac{-1}{2} \ln| 6 - \sqrt{36-4} | - F(2) , \) where the values must be substituted numerically.

Determine the Value of C

From the graph, ensure \( F(6) - F(2) = 0 \) by adding a constant \(C\). The integration over a symmetric interval can exhibit that the constant \(C\) simply modifies the initial condition. When computing with \(C = 0\), the result reflects correctly the absolute definite integral in changes indicated by the graph.

Key concepts

Definite Integral

When we talk about the definite integral, we are referring to the integration process that calculates the net area under a curve within a specified interval, in our case from \(a = 2\) to \(b = 6\). In simpler terms, a definite integral is just a number representing all the accumulated quantities from a function \(f(x)\) across this interval, precisely like summing slices of areas under \(f(x)\).

The key property is that definite integrals yield a real number, unlike indefinite integrals which produce a function. It can also be thought of as the limit of sums, particularly when we have functions that constantly change. This capacity for measurement explains their use in applications everywhere from physics to economics: measuring distances, areas, and volumes, among other things.

To compute, we employ the Fundamental Theorem of Calculus, which links differentiation and integration. We perform the calculation by:

• Finding an antiderivative \(F(x)\) of the integrand function.
• Evaluating \(F(b) - F(a)\).

This difference provides the value of the definite integral, representing the total "accumulation" over the interval \([a, b]\). In the context provided, this gave us the change from \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | \) at \(x = 2\) to \(x = 6\).

Integration Techniques

Integration techniques can be thought of as diverse tools in our mathematical toolkit, aiding in finding antiderivatives or computing integrals when straightforward methods don't apply. The integral given \(\int \frac{1}{x \sqrt{x^{2}-4}} \, dx \) is a classic example that benefits from these techniques. Rather than manually performing this brute-force integration, recognizing it as a standard form is crucial.

Integration can be sophisticated, especially when functions involve elements like roots or composite forms such as products and quotients. Frequently, methods like substitution, integration by parts, or partial fractions can simplify otherwise daunting integrals.

• Substitution is often useful when there's a composite function, replacing parts with simpler variables to work with.
• Integration by parts helps when tackling products of functions, drawing on differentiation techniques.
• Partial fraction decomposition is helpful for fractions where the denominator can be split into simpler polynomial parts.

In our example, the function is neatly solved by recognizing its relation to the derivative of the inverse hyperbolic secant, giving us back variables in intuitively understood forms. \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | + C \). Understanding these methods not only saves time but also develops deeper comprehension for tackling unique integral challenges.

Graphing Calculator

Graphing calculators represent a powerful tool for visually interpreting and solving complex functions like the integral from our exercise. By plotting functions such as \( F(x) = \frac{-1}{2} \ln| x - \sqrt{x^{2}-4} | \), we can easily verify solutions, find constants, or explore function behaviors within specific intervals.

These devices help bridge the gap between abstract integral concepts and tangible graphical intuitions. Here’s how to effectively utilize them:

• Input the antiderivative expression correctly into the calculator's function or graph mode.
• Adjust the window interval settings to ensure the interval \([2, 6]\) is covered, providing a proper view of the desired graph section.
• Use the calculator's functionality to trace along the curve, gathering approximate values at specific points like \(x = 2\) and \(x = 6\).
• Explore settings to analyze aspects like slope or intercepts, giving deeper insights especially when you adjust the constant \(C\).

A graphing calculator lets you see the immediate impact of changing parameters, like adding constants, which complements theoretical calculus skills with instant visuals and interactive feedback.