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] \(\int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} d x\) over \([0,2]\)

Short answer

Graph the antiderivative over [0,2] and adjust constant C to match the definite integral.

Step-by-step solution

Identify the function to integrate

We are given the function \(f(x) = \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}}\) and we need to find its antiderivative over the interval \([0, 2]\).

Compute the Antiderivative

Use a calculator or symbolic integration tool to evaluate the indefinite integral \( \int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} \, dx\) to find an expression for its antiderivative, \(F(x)\). This might involve complex expressions not easily written by hand.

Evaluate the Definite Integral Numerically

Use a calculator to evaluate \( \int_{0}^{2} \frac{2 e^{-2 t}}{\sqrt{1-e^{-4 t}}} \, dt \) to approximate the value of the definite integral over \([0, 2]\).

Graph the Antiderivative

Using graphing technology or a calculator, graph the antiderivative \( F(x) \) over \([0, 2]\). Ensure \(C = 0\) initially to understand the general shape of the function.

Determine Value of C

Adjust the constant \(C\) in the expression of \(F(x)\) such that when evaluated at \(x=2\), it aligns with the value obtained from the definite integral calculated in Step 3. Adjust \(C\) to "shift" the antiderivative to fit the definite integral value.

Key concepts

Antiderivative

In integral calculus, an antiderivative is a function whose derivative is the original function we started with. This means if we differentiate our antiderivative, we would get back the original function. This process is essentially the reverse of differentiation. To find the antiderivative of a function, we integrate it, which includes finding an indefinite integral. When calculating an indefinite integral, the result is expressed as a function plus a constant, typically denoted as "C." This "C" represents any constant value that could potentially be added to our function since the derivative of any constant is zero.For example, if we have a function like:\[ \int \frac{2 e^{-2x}}{\sqrt{1-e^{-4x}}} \, dx = F(x) + C \]The function \(F(x)\) is called the antiderivative. What makes finding the precise formula for an antiderivative somewhat challenging is that it often involves complex techniques and calculations. However, using calculators or integration tools can simplify this task greatly.

Definite Integral

A definite integral is used to calculate the total accumulation of a function over a set interval, typically presented as \([a, b]\). It gives a numerical value that represents the area under the curve of the function from point \(a\) to point \(b\). Essentially, the definite integral is about summing up infinitesimally small pieces of the function across an interval.The notation for a definite integral is:\[ \int_{a}^{b} f(t) \, dt \]This provides a specific value, unlike the indefinite integral, which includes a constant "C." In the exercise given, we were to find the definite integral of the function:\[ \int_{0}^{2} \frac{2 e^{-2t}}{\sqrt{1-e^{-4t}}} \, dt \]This value can be approximated using numerical methods on a calculator when the integration is overly complex. Adjusting the constant "C" in the antiderivative allows us to match the antiderivative's value at the upper limit of the interval with the calculated definite integral.

Graphing Technology

Graphing technology is an invaluable tool in understanding the behavior of functions, especially when dealing with complex expressions like those found in calculus. It allows you to visualize the shape and behavior of a function or its antiderivative over a specified interval.By graphing the antiderivative, students can observe how it behaves across the interval \([0, 2]\) in our given exercise. This process involves initially setting the constant \(C\) to zero to see the basic shape of the function. After plotting, one can adjust the constant \(C\) interactively to fit any required conditions, such as aligning the endpoint of the antiderivative's graph to match the definite integral's value at \(x=2\). This visual and interactive approach deepens understanding by bridging the gap between abstract mathematical concepts and concrete visualization.Using graphing technology like graphing calculators or software helps optimize the function analysis, allowing students to identify trends and patterns easily without manually calculating numerous data points. Such tools enhance learning by providing clear insights into the dynamics of mathematical functions.