/*! 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 8 Verify the statement by showing ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 8

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C $$

Short Answer

Expert verified
Indeed, the derivative of the right side, when simplified, is equated to the integrand of the left side. Hence, the statement has been verified.

Step by step solution

01

Differentiate the right-hand side

Begin by differentiating the right-hand side of the equation, which is \(\frac{2(x^{2}+3)}{3\sqrt{x}}+C\). Use the quotient and chain rules for differentiation.
02

Quotient Rule

The quotient rule states that, for two functions u and v, where v 鈮� 0, their derivative is: \((u/v)'=\left(vu'-uv'\right)/v^{2}\). Set \(u=2(x^{2}+3)\) and \(v=3x^{1/2}\), then find \(u'\) and \(v'\). \(u'=4x\) and \(v'=3/2x^{-1/2}\). Substituting into the quotient rule
03

Simplify the Expression

Upon substituting \(u\), \(u'\), \(v\) and \(v'\) into the quotient rule and simplifying, the result should be the integrand function on the left-hand side of the original equation, viz. \((x^{2}-1)/x^{3 / 2}\)

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

Two models, \(R_{1}\) and \(R_{2}\), are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with \(t=7\) corresponding to \(2007 .\) Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period? $$ R_{1}=7.21+0.58 t, R_{2}=7.21+0.45 t $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=4 y-y^{2}, \quad[0,4] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=e^{0.5 x}, g(x)=-\frac{1}{x}, x=1, x=2 $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=3 x^{2}+1 \quad[-1,3] $$

Lorenz Curve Economists use Lorenz curves to illustrate the distribution of income in a country. Letting \(x\) represent the percent of families in a country and \(y\) the percent of total income, the model \(y=x\) would represent a country in which each family had the same income. The Lorenz curve, \(y=f(x)\), represents the actual income distribution. The area between these two models, for
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

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.