/*! 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 10 Evaluate the definite integral o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 10

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{-2}^{-1}\left(u-\frac{1}{u^{2}}\right) d u $$

Short Answer

Expert verified
The definite integral of the given function from -2 to -1 is \(-1/2\).

Step by step solution

01

Split the Integral

Rewrite the integral as the sum of two integrals: \[\int_{-2}^{-1}\left(u-\frac{1}{u^{2}}\right) du = \int_{-2}^{-1} u du - \int_{-2}^{-1}\frac{1}{u^{2}} du\] This simplifies the problem by breaking it into two simpler integrals.
02

Antiderivative of the First Integral

Find the antiderivative of the first integral using the power rule of integration:Given an integral of the form \(\int x^n dx, n \ne -1\), the antiderivative is \(\frac{1}{n+1}x^{n+1}\) + C.So, the antiderivative of \(u\) is \(\frac{1}{2}u^{2}+C_1 \).
03

Antiderivative of the Second Integral

Evaluate the antiderivative of the second integral using the power rule. Given that \( \frac{1}{u^2}=\frac{u^{-2}}{1} \) , let \(n=-2\), then:\(\frac{1}{n+1}u^{n+1} = -u^{-1} + C_2\)The antiderivative of \( \frac{1}{u^2} \) is \(-u^{-1}\) + \(C_2\).
04

Applying Limits to the Antiderivatives

Now substitute the limits into the antiderivatives, applying the fundamental theorem of calculus:The integral from -2 to -1 can be calculated as \(F(-1) - F(-2)\), where \(F(u) = [ \frac{1}{2}u^2 - u^(-1) ]\):\[\int_{-2}^{-1} u du = \frac{1}{2}(-1)^2 - (-1)^{-1} - [\frac{1}{2}(-2)^2 - (-2)^{-1}]\]
05

Calculate the Result

Calculate the result based on above expression:\[= \frac{1}{2} - 1 - [\frac{-1}{2} + 1] = 1 - \frac{3}{2} = -\frac{1}{2}\]

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

In Exercises \(47-52,\) evaluate the integral. \(\int_{0}^{\ln 2} \tanh x d x\)

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{\sin x} \sqrt{t} d t $$

Use the equation of the tractrix \(y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}, \quad a>0\) Let \(L\) be the tangent line to the tractrix at the point \(P .\) If \(L\) intersects the \(y\) -axis at the point \(Q\), show that the distance between \(P\) and \(Q\) is \(a\).

Find the integral. \(\int \frac{\cosh x}{\sqrt{9-\sinh ^{2} x}} d x\)

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{1}^{x} \frac{1}{t} d t $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.