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Chapter 1: Problem 420

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

Short Answer

Expert verified
Calculate the indefinite integral and add a constant \(C\) that matches the definite integral result at \(x=2\).

Step by step solution

01

Understand the Problem

We are asked to find the anti-derivative of the function \( f(x) = \frac{2 e^{-2x}}{\sqrt{1-e^{-4x}}} \) over the interval \([0, 2]\) and determine a constant \(C\) that, when added to the indefinite integral, yields the same value as the definite integral \( F(x) = \int_{0}^{x} f(t) \, dt \).
02

Compute Indefinite Integral

Using a calculator (such as a graphing calculator or software), find the indefinite integral \( \int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} \, dx \). This represents the anti-derivative of \( f(x) \) with respect to \(x\).
03

Graph the Indefinite Integral on given Interval

Graph the solution found in Step 2 over the interval \([0,2]\). This will visually represent the anti-derivative and help in comparing its value at specific points against the definite integral.
04

Compute Definite Integral from a to b

Calculate the definite integral \( F(2) = \int_{0}^{2} \frac{2 e^{-2 t}}{\sqrt{1-e^{-4 t}}} \, dt \) using the calculator. This value represents the accumulation of area under the curve of \(f(x)\) from \(x=0\) to \(x=2\).
05

Determine Value of C

Compare the values obtained from the indefinite integral at \(x=2\) with the result of the definite integral. The difference between these two should give the constant \( C \). So, solve \( F(2) = \int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} \, dx + C \) at \(x=2\).
06

Approximate Value of C

Using the values from Step 4 and the graph from Step 3, solve for \( C \) such that the computed indefinite integral plus \( C \) equals the definite integral \( F(2) \).

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Key Concepts

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

Definite Integral
A definite integral is one of the fundamental concepts of calculus. It represents the accumulation of a function's values over an interval. In simple terms, imagine it as the total area under a curve from one point to another on the x-axis.
For example, you can calculate it for the function from x = a to x = b, written as \( \int_a^b f(x) \, dx \). This takes into consideration the exact limits provided, meaning it gives us a number rather than a function as a result.
  • Used to calculate area under a curve.
  • Provides a precise numerical answer.
  • Represents the accumulation of quantities, such as distance or volume, over a certain interval.
Exploring the definite integral can lead to solving real-world problems where these accumulations matter, ensuring we understand change not just instantaneously, but over intervals.
Indefinite Integral
The indefinite integral, unlike the definite one, provides a general solution to find the antiderivative of a function. It returns a function plus a constant, commonly symbolized as \(C\), because there are infinitely many antiderivatives for any given function.
Represented as \( \int f(x) \, dx \), the indefinite integral emphasizes the function form, showcasing the broader picture without bounds.
  • Yields a function rather than a number.
  • Includes a constant of integration \( C \).
  • Reflects all possible antiderivatives a function could have.
This concept helps students grasp the general solution approach, which can later be tweaked or manipulated to fit specific scenarios, such as equations with initial conditions.
Graphing Calculators
Graphing calculators are powerful tools in mathematics, especially for students learning calculus. They help visualize functions, graph their antiderivatives, and even compute numerical integrals, both definite and indefinite.
Using a graphing calculator allows you to:
  • Plot detailed graphs to understand function behavior over specific intervals.
  • Calculate integrals swiftly, which is crucial when complex functions make manual calculations difficult.
  • Identify intersections and analyze changes in graphs to find constants like \(C\) in integrals.
For those tackling calculus concepts, mastering graphing calculator functionalities can enhance understanding and ensure accurate mathematical computations, making them invaluable throughout your calculus journey.
Constant of Integration
The constant of integration \( C \) appears whenever we determine an indefinite integral. Since differentiation is an operation that loses information about constants, when computing antiderivatives, \( C \) represents all those lost constants that could exist.
This constant is critical because:
  • It reflects all possible vertical shifts of an antiderivative graph.
  • Allows personalization of solutions to meet specific, external conditions, often initial conditions.
  • Acts as a bridge between indefinite and definite integrals, as seen in finding a specific \( C \) that aligns with given interval evaluations.
In practice, the constant of integration ensures that solutions to differential equations or other calculus problems remain flexible yet precise when extra conditions are provided. It's a reminder of the infinite solutions calculus often reveals.

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Most popular questions from this chapter

If \(f\) is 1-periodic and \(\int_{0}^{1} f(t) d t=A,\) is it necessarily true that \(\int_{a}^{1+a} f(t) d t=A\) for all \(A\) ?

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. $$ [\mathbf{T}] y=x\left(1-x^{2}\right)^{3} \text { over }[-1,2] $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x $$

Show that the average value of \(\sin ^{2} t\) over \([0,2 \pi]\) is equal to \(1 / 2\) Without further calculation, determine whether the average value of \(\sin ^{2} t\) over \([0, \pi]\) is also equal to \(1 / 2\).

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t^{2} \cos ^{2}\left(t^{3}\right) \sin \left(t^{3}\right) d t $$
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