/*! 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 7 Evaluate the integral. \(\int_... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 7

Evaluate the integral. \(\int_{-1}^{0}\left(2 x-e^{x}\right) d x\)

Short Answer

Expert verified
The definite integral evaluates to \(-2 + \frac{1}{e}\).

Step by step solution

01

Set Up the Integral

First, we need to evaluate the definite integral \(\int_{-1}^{0} (2x - e^x) \, dx\). This means we need to find the antiderivative of the function \(f(x) = 2x - e^x\) and then use the Fundamental Theorem of Calculus to calculate the definite integral from \(-1\) to \(0\).
02

Find the Antiderivative

To find the antiderivative of \(f(x) = 2x - e^x\), we integrate each term separately:- The antiderivative of \(2x\) is \(x^2\).- The antiderivative of \(-e^x\) is \(-e^x\).Therefore, the antiderivative \(F(x)\) of \(f(x)\) is \(F(x) = x^2 - e^x + C\), where \(C\) is the constant of integration.
03

Apply the Fundamental Theorem of Calculus

Now, we apply the Fundamental Theorem of Calculus, which tells us that: \[\int_{-1}^{0} (2x - e^x) \, dx = F(0) - F(-1)\]Substitute the values into the antiderivative:- \(F(0) = (0)^2 - e^{0} = -1\)- \(F(-1) = (-1)^2 - e^{-1} = 1 - \frac{1}{e}\) Thus, \[\int_{-1}^{0} (2x - e^x) \, dx = (-1) - \left(1 - \frac{1}{e}\right)\]
04

Simplify the Result

Simplify the expression:\[(-1) - \left(1 - \frac{1}{e}\right) = -1 - 1 + \frac{1}{e} = -2 + \frac{1}{e}\]Thus, the value of the integral is \(-2 + \frac{1}{e}\).

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!

Key Concepts

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

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a cornerstone in understanding definite integrals. It bridges the world of differentiation and integration, providing a way to evaluate definite integrals through antiderivatives. In essence, the theorem consists of two main parts:
  • The first part of the theorem tells us that if you have a continuous function and a point within its interval, you can express the accumulation function as an antiderivative.
  • The second part lets us evaluate a definite integral by using antiderivatives. Specifically, if you have an antiderivative \( F(x) \) of \( f(x) \), then the definite integral from \( a \) to \( b \) is \( F(b) - F(a) \).
In the exercise provided, we use the second part of the theorem. By finding the antiderivative of the function \( 2x - e^x \) first, we are able to calculate the definite integral by evaluating the antiderivative at the upper and lower bounds \( 0 \) and \(-1\), respectively.
Antiderivatives
Antiderivatives are crucial when solving integrals, as they are essentially the reverse process of differentiation. To find an antiderivative, you look for a function whose derivative gives you the original function you started with. For example, the function \( f(x) = 2x \) has an antiderivative of \( x^2 \) because the derivative of \( x^2 \) is \( 2x \).
  • The antiderivative of \( e^x \) is \( e^x \) itself since its derivative remains the same. Thus, the expression \( -e^x \) has an antiderivative of \( -e^x \).
  • So overall, the antiderivative of \( 2x - e^x \) is \( x^2 - e^x \).
Bear in mind, when calculating indefinite integrals, we usually add a constant of integration \( C \). However, in the context of a definite integral using the theorem, this constant cancels out.
Exponential Functions
Exponential functions are unique mathematical expressions where the variable is in the exponent. The simplest exponential function is \( e^x \), where \( e \) is approximately equal to 2.71828, known as Euler's number. This type of function grows rapidly and is its own derivative:
  • The differentiation or the derivative of \( e^x \) is \( e^x \), making it very special and powerful in calculus.
  • Hence, its antiderivative is also \( e^x \).
In our example, \(-e^x\) is a term in the integrand, and finding its antiderivative involves no complexity because it simply remains \(-e^x\). In the evaluation of the integral from \(-1\) to \(0\), this characteristic helps to directly compute the integral without additional steps. It's a demonstration of how beautifully simple yet profound exponential functions are in calculus.

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

$$\begin{array}{c}{\text { If a factor of the denominator is an irreducible quadratic, }} \\ {\text { such as } x^{2}+1, \text { the corresponding partial fraction has a }} \\ {\text { linear numerator. For instance, }} \\\ {f(x)=\frac{2 x^{2}+x+1}{x\left(x^{2}+1\right)}=\frac{A}{x}+\frac{B x+C}{x^{2}+1}}\end{array}$$ $$\begin{array}{l}{\text { Determine the values of } A, B \text { , and } C \text { and use them to }} \\ {\text { evaluate } f f(x) d x .}\end{array}$$

Find the general indefinite integral. \(\int \sec t(\sec t+\tan t) d t\)

If \(f(1)=12, f^{\prime}\) is continuous, and \(\int_{1}^{4} f^{\prime}(x) d x=17,\) what is the value of \(f(4) ?\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{0}^{\infty} \frac{e^{x}}{e^{2 x}+3} d x\)

If oil leaks from a tank at a rate of \(r(t)\) gallons per minute at time \(t,\) what does \(\int_{0}^{120} r(t) d t\) represent?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics 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.