Problem 410 - Calculus Volume 1

Question

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{e^{x}}{1+e^{2 x}} d x \text { over }[-6,6]$$

Step-by-Step Solution

Answer

The antiderivative is \(\frac{1}{2} \ln |1 + e^{2x}|\), and \(C=0\).

Step 1: Find the Antiderivative

We need to find the antiderivative \( \int \frac{e^{x}}{1+e^{2x}} \, dx \). This can be simplified using substitution. Set \( u = 1 + e^{2x} \), then \( du = 2e^{2x} \, dx \). Notice that \( e^{x} = \frac{du}{2e^{x}} \), so \( dx = \frac{du}{2e^{x}} \). Substituting gives: \[ \int \frac{e^{x}}{u} \cdot \frac{du}{2e^{x}} = \int \frac{1}{2u} \, du \ = \frac{1}{2} \int \frac{1}{u} \, du \ = \frac{1}{2} \ln |u| + C \]Substituting back for \( u \) gives: \[ \frac{1}{2} \ln |1 + e^{2x}| + C \]

Step 2: Graph f(x) and Its Antiderivative

Using a graphing calculator, plot \( f(x) = \frac{e^{x}}{1 + e^{2x}} \) over the interval \([-6, 6]\). Also, graph the antiderivative \( F(x) = \frac{1}{2} \ln |1 + e^{2x}| \) over the same interval. Ensure that the constant \( C = 0 \) initially when graphing the antiderivative.

Step 3: Identify C for the Definite Integral

The definite integral form is \( F(x) = \int_{-6}^{x} \frac{e^{t}}{1 + e^{2t}} \, dt \). To determine the constant \( C \) such that we integrate correctly from \([-6, 6]\), calculate \( F(-6) \). At \( x = -6 \), using the antiderivative gives: \[ F(-6) = \frac{1}{2} \ln |1 + e^{-12}| \approx 0 \]Therefore, if \( F(x) \) must recover the definite integral starting from \(-6\), the constant \( C \) remains zero since it naturally evaluates to zero.

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, representing the accumulation of quantities, such as areas under curves. These are expressed in the form \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the function to be integrated, and \( a \) and \( b \) are the limits of integration. Definite integrals have a geometric interpretation as the net area between the function and the x-axis across the interval from \( a \) to \( b \).

In the given exercise, we're asked to find the antiderivative and then recover the definite integral from a specific point, which generally involves finding a constant \( C \). This constant ensures that when integrating from the lower limit, the antiderivative aligns with the area under the curve as initially defined by the definite integral.

• Understand that the definite integral provides both a value and a boundary-defined area.
• It translates into real-world scenarios such as total distance traveled or total revenue generated over a time period.

Substitution Method

The substitution method is a powerful technique in calculus used to simplify integrals. It's particularly useful when dealing with complex expressions that can be made simpler with an appropriate change of variables.

In our problem, the substitution technique was applied to the integral \( \int \frac{e^{x}}{1+e^{2x}} \, dx \). Here, we set \( u = 1 + e^{2x} \), which makes the integral more manageable by reducing it to a basic form.

• This method involves replacing parts of the integral with a new variable \( u \), then deriving \( du \) from the differential of \( u \).
• Apply the technique by expressing \( dx \) in terms of \( du \), substituting back after integrating if needed.
• In this example, once integrated, we revert the process by substituting \( u \) back in terms of \( x \), to complete the antiderivative expression.

Graphing Calculators

Graphing calculators are invaluable tools for visualizing mathematical functions and their derivatives or antiderivatives. They help students better understand the behavior of functions over a specified interval by plotting them graphically.

In this problem, the exercise involves using a graphing calculator to plot the function \( f(x) = \frac{e^{x}}{1+e^{2x}} \) and its antiderivative \( F(x) = \frac{1}{2} \ln |1+e^{2x}| \) across the interval \([-6,6]\).

• Graphing calculators allow for experimenting with different values and adjustments to parameters, enhancing learning about both the function and its integral properties.
• They provide a visual confirmation of the algebraic steps involved and help in comparing the original function with its antiderivative.
• Software alternatives, such as Desmos or GeoGebra, can also be used for similar purposes, offering an interactive experience on more accessible platforms.

Calculus Step-by-Step Solutions

The step-by-step solution approach in calculus is essential for understanding complex problems by breaking them down into manageable parts. This methodical progression allows students to grasp each component and its contribution to solving the overall problem.

For the integral exercise at hand, the solution progresses methodically:

- **Step 1**: Simplify using substitution, finding the antiderivative.
- **Step 2**: Visualize with a graphing calculator, plotting the function and its antiderivative.
- **Step 3**: Determine the constant \( C \) by calculating at a specific endpoint. This is crucial for ensuring the definite integral returns expected area values.

This structured format enables learners to focus on the logic behind each step, deepening their understanding of integration, especially when multiple techniques like substitution are involved. Emphasizing each step allows for a solid grasp of fundamental calculus principles and aides in translating exercises into comprehensible computations.