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Chapter 8: Problem 27

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{\left(1-x^{2}\right)^{3 / 2}}{x^{6}} d x $$

Short Answer

Expert verified
\(-\left[\frac{1}{5x^5} - \frac{2}{3x^3} + \frac{1}{x}\right] + C\)."

Step by step solution

01

Identify the Trigonometric Substitution

Recognize that the expression \(1-x^2\) suggests a trigonometric substitution. We will use the substitution \(x = \sin(\theta)\), which consequently means \(dx = \cos(\theta) d\theta\). This substitution simplifies the expression \(1 - x^2\) to \(\cos^2(\theta)\).
02

Substitute and Simplify the Integral

Substitute \(x = \sin(\theta)\) and \(dx = \cos(\theta) d\theta\) into the integral. The integral becomes: \[\int \frac{(1 - \sin^2(\theta))^{3/2}}{\sin^6(\theta)} \cos(\theta) \, d\theta.\] This simplifies to: \[\int \frac{\cos^3(\theta)}{\sin^6(\theta)} \cos(\theta) \, d\theta = \int \frac{\cos^4(\theta)}{\sin^6(\theta)} \, d\theta.\]
03

Convert to Tangent and Secant

Recognize that \(\frac{\cos^4(\theta)}{\sin^6(\theta)}\) can be rewritten in terms of tangent and secant: \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Substitute: \[\int \cot^4(\theta) \csc^2(\theta) \, d\theta.\]
04

Use Trigonometric Identities to Simplify Further

Use a substitution, \(u = \csc(\theta)\), so that \(du = -\csc(\theta)\cot(\theta)\,d\theta\). This makes the integral: \[\int \cot^4(\theta) \csc^2(\theta) \, d\theta = \int \csc^4(\theta) - 2\csc^2(\theta) + 1 \, d\theta,\] and further using substitution, we get \( -\int (u^4 - 2u^2 + 1) du\).
05

Integrate and Simplify Back

Integrate the expression in terms of \(u\): \(-\int (u^4 - 2u^2 + 1) \, du\) gives \(-\left[\frac{u^5}{5} - \frac{2u^3}{3} + u\right] + C\). Replacing \(u = \csc(\theta)\), convert back to \(x\) using the substitution \(x = \sin(\theta)\): \(\csc(\theta) = \frac{1}{x}\). Thus, \( -\left[\frac{1}{5x^5} - \frac{2}{3x^3} + \frac{1}{x}\right] + C.\)

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Key Concepts

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

Integral Calculus
Integral Calculus is a branch of mathematics that deals with integrals and their applications. Integrals can be thought of as the reverse process of derivatives. They help us find areas under curves and solve problems related to accumulation.Integrals are classified into two types:
  • Definite Integrals: These provide a numerical value representing the area under a curve between two points. The notation for definite integrals is \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.
  • Indefinite Integrals: These represent a family of functions and are written without limits. The notation is \( \int f(x) \, dx \), and these yield a general anti-derivative of \( f(x) \).
Integral Calculus is powerful in modeling a wide range of phenomena, from physics to finance. In the specific problem we examined, we used a form of indefinite integral to simplify and find the solution.Understanding how to work with integrals is essential for solving many advanced mathematical problems.
Trigonometric Identities
Trigonometric Identities are equations involving trigonometric functions that hold true for all values of the variable. They are crucial tools in simplifying and solving integrals involving trigonometric functions.Some essential trigonometric identities are:
  • Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • Reciprocal Identities: For example, \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
  • Quotient Identities: For example, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
  • Double Angle Identities: Such as \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
In our problem, these identities helped us transform complex expressions into simpler forms. Recognizing patterns and applying the correct identities is key to solving integrals efficiently.
Mathematical Substitution
Mathematical Substitution is a method used to simplify integrals by changing variables. This technique is especially useful when a direct integration seems complicated or unmanageable.When using substitution:
  • Identify a suitable substitution that transforms the original variable into a new variable. In our example, we used \( x = \sin(\theta) \).
  • Express differential elements in terms of the new variable, such as \( dx = \cos(\theta)\, d\theta \).
  • Rewrite the entire integral in terms of this new variable, which often simplifies the expression significantly.
Substitution is a powerful strategy in calculus that allows us to handle tricky integrals by reducing them to a more manageable form. Mastering substitution can make a huge difference in your ability to solve integral calculus problems.
Definite Integrals
Definite Integrals are used to calculate the area under a curve over an interval \([a, b]\). Unlike indefinite integrals, they yield a single number as the result.The process of finding definite integrals:
  • Determine the antiderivative of the function.
  • Evaluate the antiderivative at the upper limit \( b \) and lower limit \( a \).
  • Subtract the value of the antiderivative at \( a \) from its value at \( b \): \( F(b) - F(a) \).
Definite integrals have many applications:
  • In physics, they can calculate distances, work, and other quantities.
  • In probability, they help find probability distributions.
  • In economics, they can determine consumer and producer surplus.
Even though our particular problem focused on indefinite integrals, a solid understanding of definite integrals is essential for advancing in calculus.

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Most popular questions from this chapter

Manufacturing time The assembly time in minutes for a component at an electronic manufacturing plant is normally distributed with a mean of \(\mu=55\) and standard deviation \(\sigma=4 .\) What is the probability that a component will be made in less than one hour?

Blood pressure Diastolic blood pressure in adults is normally distributed with \(\mu=80 \mathrm{mm}\) Hg and \(\sigma=12 \mathrm{mm}\) Hg. In a random sample of 300 adults, how many would be expected to have a diastolic blood pressure below 70 \(\mathrm{mm}\) Hg?

Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year's rainfall will exceed 17 in.?

Spacerraft components A component of a spacecraft has both a main system and a backup system operating throughout a flight. The probability that both systems fail sometime during the flight is \(0.0148 .\) Assuming that each system separately has the same failure rate, what is the probability that the main system fails during the flight?

The infinite paint can or Gabriel's horn As Example 3 shows, the integral \(\int_{1}^{\infty}(d x / x)\) diverges. This means that the integral $$\int_{1}^{\infty} 2 \pi \frac{1}{x} \sqrt{1+\frac{1}{x^{4}}} d x$$ which measures the surface area of the solid of revolution traced out by revolving the curve \(y=1 / x, 1 \leq x,\) about the \(x\) -axis, diverges also. By comparing the two integrals, we see that, for every finite value \(b>1\) . $$\int_{1}^{b} 2 \pi \frac{1}{x} \sqrt{1+\frac{1}{x^{4}}} d x>2 \pi \int_{1}^{b} \frac{1}{x} d x$$ However, the integral $$\int_{1}^{\infty} \pi\left(\frac{1}{x}\right)^{2} d x$$ for the volume of the solid converges. \begin{equation} \begin{array}{l}{\text { a. Calculate it. }} \\ {\text { b. This solid of revolution is sometimes described as a can that }} \\ {\text { does not hold enough paint to cover its own interior. Think }} \\ {\text { about that for a moment. It is common sense that a finite }}\\\ {\text { amount of paint cannot cover an infinite surface. But if we fill }} \\\ {\text { the horn with paint (a finite amount), then we will have cov- }} \\\ {\text { ered an infinite surface. Explain the apparent contradiction. }}\end{array} \end{equation}
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