/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 16

Evaluate the following integrals. $$\int\left(36-9 x^{2}\right)^{-3 / 2} d x$$

Short Answer

Expert verified
Question: Evaluate the integral of the function \((36 - 9x^2)^{-3/2}\). Answer: $$-\frac{1}{6}(36 - 9x^2)^{-1} + C$$

Step by step solution

01

Identify the substitution variable and derivative

For this integral, we can use the substitution method: u = \(6 - 3x^2\) First, we'll differentiate u with respect to x: \(du = -6x \, dx\) Now we'll solve for \(dx\): \(dx = \frac{du}{-6x}\)
02

Replace in the integral

Now we can replace the function in the integral with the variable u: $$\int(36 - 9x^2)^{-3/2}dx = \int u^{-3/2} \frac{du}{-6x}$$ We can see that we are missing the factor \(-6x\) in our substitution. To get this, we differentiate the original substitution: \(u = 6 - 3x^2\) Then, solve for x: \(x^2 = 2 - \frac{u}{3}\) \(x = \sqrt{2 - \frac{u}{3}}\) Now differentiate x with respect to u: \(\frac{dx}{du} = -\frac{1}{6}u^{-1/2}\) Now, we substitute x and dx in the integral: $$\int (36 - 9x^2)^{-3/2} dx =\int u^{-3/2} \, (-\frac{1}{6}u^{-1/2}du) $$
03

Integrate

Now we integrate the new function with respect to u: $$\int (-\frac{1}{6}u^{-2}du)$$ Using the power rule for integration, we get: $$-\frac{1}{6} \int u^{-2}du =-\frac{1}{6}(u^{-1})$$ Now substitute back in terms of x: $$-\frac{1}{6}(36 - 9x^2)^{-1}$$ Add the constant of integration C to complete the solution: $$-\frac{1}{6}(36 - 9x^2)^{-1} + C$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Challenge Show that with the change of variables \(u=\sqrt{\tan x}\) the integral \(\int \sqrt{\tan x} d x\) can be converted to an integral amenabl to partial fractions. Evaluate \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\)

The heights of U.S. men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. This means that the fraction of men with a height between \(a\) and \(b\) (with \(a

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{3 x^{2}+4 x-6}{x^{2}-3 x+2} d x$$

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=t \longrightarrow F(s)=\frac{1}{s^{2}}$$

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(y\) -axis on the interval (-1,1] is revolved about the line \(x=-1.\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.