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Chapter 7: Problem 17

Calculate. $$\int \frac{x}{3-x^{2}} d x$$

Short Answer

Expert verified
The short answer to the given question would be: \(\int \frac{x}{3-x^{2}} dx = -\frac{\sqrt{3}}{2}\ln\left|\sqrt{3} + x\right| + \frac{\sqrt{3}}{2}\ln\left|\sqrt{3} - x\right| + C\), where \(C\) is the constant of integration.

Step by step solution

01

Factor the denominator

First, we factor the denominator. We have: \(3-x^{2} = (\sqrt{3}+x)(\sqrt{3}-x)\)
02

Perform partial fraction decomposition

We decompose the integrand into partial fractions. Let: \(\frac{x}{3-x^{2}} = \frac{A}{\sqrt{3}+x} + \frac{B}{\sqrt{3}-x}\) Multiplying both sides by \((\sqrt{3}-x)(\sqrt{3}+x)\), we get: \(x = A(\sqrt{3}-x) + B(\sqrt{3}+x)\) To solve for the constants \(A\) and \(B\), we equate the coefficients of the powers of \(x\). Coefficients of constant terms: \((A + B)\times \sqrt{3} = 0\) From this equation, we can find that \(A = -B\). Coefficients of \(x\): \(-A + B = 1\) Now solve for A and B: \begin{align*} A + (-A) &= -\sqrt{3}\\ 2A &= -\sqrt{3}\\ A &= -\frac{\sqrt{3}}{2}\\ B &= \frac{\sqrt{3}}{2} \end{align*}
03

Integrate the decomposed fractions

Now that we have the constants \(A\) and \(B\), we can integrate the sum of the decomposed fractions: \[\int \frac{x}{3-x^{2}} dx = \int \left(\frac{-\frac{\sqrt{3}}{2}}{\sqrt{3}+x}+\frac{\frac{\sqrt{3}}{2}}{\sqrt{3}-x} \right) dx\] \[\int \frac{x}{3-x^{2}} dx = -\frac{\sqrt{3}}{2}\int \frac{1}{\sqrt{3}+x} dx + \frac{\sqrt{3}}{2} \int \frac{1}{\sqrt{3}-x} dx\] After solving these integrals, we get: \(\int \frac{x}{3-x^{2}} dx = -\frac{\sqrt{3}}{2}\ln\left|\sqrt{3} + x\right| + \frac{\sqrt{3}}{2}\ln\left|\sqrt{3} - x\right| + C\) Here, \(C\) is the constant of integration.

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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 mathematical technique often used in Calculus to simplify complex rational expressions, particularly when integrating. This method breaks down a fraction into simpler 'partial' fractions that are easier to handle.

Consider a rational function with a numerator smaller in degree than the denominator. We can express this function as a sum of simpler fractions, each with its unique denominator that is a factor of the original denominator, and a numerator that is usually of lesser degree. These simpler fractions are what we call 'partial fractions'.

Finding Constants in Partial Fractions

To find the constants for each partial fraction, we multiply the entire equation by the original denominator to clear the fractions. Then we equate coefficients of like terms or select strategic values for the variable to solve for the unknown constants.
Indefinite Integral
An indefinite integral is the antiderivative of a function. It is a fundamental concept within integral calculus representing a class of functions associated with the area under a curve of a graphed function. An indefinite integral doesn't have specific bounds and includes an arbitrary constant, commonly denoted as C.

Notation and Interpretation

The notation \( \int f(x)dx \) represents the family of all antiderivatives of \(f(x)\). Intuitively, this process reverses differentiation, leading you from the rate of change (derivative) back to the original quantity (function). The constant of integration signifies that there are infinitely many antiderivatives, each varying by a constant.
Integrating Rational Functions
Integrating rational functions, which are ratios of polynomials, requires specific techniques depending on the characteristics of the function. When the degree of the numerator is greater than or equal to the degree of the denominator, you perform long division. Otherwise, partial fraction decomposition is typically the preferred method.

Technique Application

After decomposing into partial fractions, each simpler fraction is integrated individually. Since the results of these integrations are easier to compute, this process breaks down a potentially challenging integration problem into manageable pieces. End results will often include logarithmic or inverse trigonometric functions as part of the solution.
Calculus
Calculus is an advanced branch of mathematics that deals with continuous change. It is fundamentally divided into two areas: Differential Calculus involving derivatives and rates of change, and Integral Calculus focusing on accumulation and area under curves.

Real-World Application

Calculus is invaluable in physics, engineering, economics, statistics, and even medicine. It not only helps us solve complex mathematical problems, like those involving rational functions and their integration but also gives us the language to describe and investigate a myriad of real-world processes. It enables us to understand the dynamics of motion, the growth of patterns, and the changes in rates effectively.

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Most popular questions from this chapter

You are 45 years old and are looking forward to an annual pension of 50,000 dollar per year at age \(65 .\) What is the present day purchasing power (present value) of your pension if money can be invested over this period at a continuously compounded interest rate of: (a) 4 \(\% ?\) (b) \(6 \% ?\) (c) \(8 \% ?\)

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon- 12, denoted \(^{12} \mathrm{C}\) (a stable isotope), and carban-14, denoted \(^{14} \mathrm{C}\) (a radioactive isotope). The ratio of the amount of \(^{14} \mathrm{C}\) to the amount of \(^{12} \mathrm{C}\) is essentially constant (approximately \(1 / 10,000\) ). When an organism dies, the amount of \(^{12}\)C Present remains unchanged, but the \(^{14} \mathrm{C}\) decays at a rate proportional to the amount present with a half-life of approximately 5700 years. This change in the amount of \(^{14}\) Relative to the amount of \(^{12} \mathrm{C}\) makes it possible to estimate the time at which the organism lived. A fossil found in an archaeological dig was found to contain \(25 \%\) of the original amount of \(^{14} \mathrm{C}\). What is the approximate age of the fossil?

A person walking along a straight path at the rate of 6 feet per second is followed by a spotlight that is located 30 feet from the path. How fast is the spotlight turning at the instant the person is 50 feet past the point on the path that is closest to the spotlight?

Draw a figure that displays the graphs of both $$f(x)=\ln x \quad \text { and } \quad g(x)=\log _{3} x$$

Evaluate. $$\int_{2}^{5} \frac{d x}{9+(x-2)^{2}}$$.
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