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Chapter 7: Problem 71

Use any method to solve for \(x\). $$ \int_{2}^{x} \frac{1}{t(4-t)} d t=0.5,2

Short Answer

Expert verified
The solution is \( x = \frac{4e^2}{1+e^2} \) with \( 2 < x < 4 \).

Step by step solution

01

Simplify the Integral Expression

To solve the given definite integral \( \int_{2}^{x} \frac{1}{t(4-t)} dt = 0.5 \), we'll start by simplifying the integrand. The expression \( \frac{1}{t(4-t)} \) can be decomposed using partial fraction decomposition. This gives us: \[ \frac{1}{t(4-t)} = \frac{A}{t} + \frac{B}{4-t} \], where \( A \) and \( B \) are constants.
02

Find Partial Fraction Constants

Set the equation \( \frac{1}{t(4-t)} = \frac{A}{t} + \frac{B}{4-t} \) and solve for \( A \) and \( B \). Equating and simplifying yields: \[ 1 = A(4-t) + Bt \]. By substituting convenient values for \( t \):- Let \( t=0 \) yields \( A = \frac{1}{4} \).- Let \( t=4 \) yields \( B = \frac{1}{4} \).The fraction becomes: \[ \frac{1}{t(4-t)} = \frac{1/4}{t} + \frac{1/4}{4-t} \].
03

Integrate the Expression with Found Constants

Substitute back into the integral: \[ \int_{2}^{x} \left( \frac{1}{4t} + \frac{1}{4(4-t)} \right) dt \].Split the integral: \[ \frac{1}{4} \int_{2}^{x} \frac{1}{t} dt + \frac{1}{4} \int_{2}^{x} \frac{1}{4-t} dt \].The integrals evaluate to:- \( \frac{1}{4} \ln |t| \Big|_{2}^{x} \)- \( -\frac{1}{4} \ln |4-t| \Big|_{2}^{x} \)
04

Compute the Values for Each Integral

Evaluate the two integrals:- For \( \frac{1}{4} (\ln x - \ln 2) = \frac{1}{4} \ln \frac{x}{2} \)- For \( -\frac{1}{4} (\ln |4-x| - \ln |4-2|) = -\frac{1}{4} \ln \frac{4-x}{2} \)Combine to get the expression: \[ \frac{1}{4} \ln \frac{x}{2} - \frac{1}{4} \ln \frac{4-x}{2} = 0.5 \]
05

Solve the Logarithmic Equation

Combine the two logarithms: \[ \frac{1}{4} \left( \ln \frac{x}{2} - \ln \frac{4-x}{2} \right) = \frac{1}{4} \ln \frac{x}{4-x} = 0.5 \].Eliminate the coefficient: \[ \ln \frac{x}{4-x} = 2 \].Exponentiate both sides: \[ \frac{x}{4-x} = e^2 \] implies:- Solve for \( x \): \( x = \frac{4e^2}{1+e^2} \), where \( 2 < x < 4 \). This value satisfies the constraint.

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Key Concepts

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

Partial Fraction Decomposition
Partial Fraction Decomposition is a technique used to simplify complex rational expressions, which can make integration more approachable. The method involves breaking down a compound fraction into a sum of simpler fractions. This is particularly useful when dealing with integration, as simpler fractions have well-known integration formulas.
Let's see how this was applied in our given problem:
  • The initial expression, \(\frac{1}{t(4-t)}\), was decomposed into \(\frac{A}{t} + \frac{B}{4-t}\).
  • By equating and substituting convenient values for \(t\), the constants \(A\) and \(B\) were determined: both equaling \(\frac{1}{4}\).
This simplification by partial fractions transformed a complex problem into an integrable form. In general, once the decomposition is done, it's straightforward to integrate each of the simpler fractions separately.
Definite Integrals
Definite integrals offer a way to calculate the area under a curve between two specified limits. These integrals have both a lower and upper limit, making the evaluation straightforward once the antiderivative is determined.
  • In our problem, the integral was examined from \(t = 2\) to \(t = x\).
  • After splitting the integral using partial fraction decomposition, the definite integral was calculated for each fraction separately.
  • Specific calculations included \(\int \frac{1}{t} dt\) and \(\int \frac{1}{4-t} dt\).
The results of these definite integrals were then used to form a logarithmic equation. Definite integrals are essential in solving such problems as they not only provide the area but also help solve equations involving unknown limits as seen in this scenario.
Logarithmic Equations
After calculating the definite integrals, the solution involves solving a logarithmic equation. Logarithmic functions often appear in calculus because of their relationship with exponential growth and decay.
  • In this task, the combined result from the integrals was \(\frac{1}{4} \ln \frac{x}{4-x} = 0.5\).
  • By isolating the logarithmic term, the equation was simplified to \(\ln \frac{x}{4-x} = 2\).
  • To solve this, both sides were exponentiated to switch from a logarithmic to an algebraic equation.
This led to the equation \(\frac{x}{4-x} = e^2\), which was solved to find the value of \(x\). Logarithmic equations like this one show the application of fundamental algebraic transformations to solve calculus problems, blending both algebra and calculus concepts elegantly.

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Most popular questions from this chapter

What is "improper" about a definite integral over an interval on which the integrand has an infinite discontinuity? Explain why Definition \(5.5 .1\) for \(\int_{a}^{b} f(x) d x\) fails if the graph of \(f\) has a vertical asymptote at \(x=a\). Discuss a strategy for assigning a value to \(\int_{a}^{b} f(x) d x\) in this circumstance.

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{d x}{x^{1 / 2}-x^{1 / 3}} $$

Medication can be administered to a patient using a variety of methods. For a given method, let \(c(t)\) denote the concentration of medication in the patient's bloodstream (measured in \(\mathrm{mg} / \mathrm{L}) t\) hours after the dose is given. The area under the curve \(c=c(t)\) over the time interval \([0,+\infty)\) indicates the "availability" of the medication for the patient's body. Determine which method provides the greater availability. Method \(1: c_{1}(t)=5\left(e^{-0.2 t}-e^{-t}\right)\) Method 2: \(c_{2}(t)=4\left(e^{-0.2 t}-e^{-3 t}\right)\)

The region bounded below by the \(x\) -axis and above by the portion of \(y=\sin x\) from \(x=0\) to \(x=\pi\) is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Find the volume of the solid that results when the region enclosed by \(y=\cos x, y=\sin x, x=0\), and \(x=\pi / 4\) is revolved about the \(x\) -axis.
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