/*! 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 3 Determine these indefinite integ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Determine these indefinite integrals. $$\int 2 \, d x$$

Short Answer

Expert verified
The integral is \( 2x + C. \)

Step by step solution

01

- Understand the Integral

Recognize that the integral \(\int 2 \, dx\) involves integrating a constant function.
02

- Apply the Integration Rule for Constants

When integrating a constant, use the rule \[\int c \, dx = cx + C \] where \( c \) is a constant and \( C \) is the constant of integration.
03

- Substitute the Values

Here, \( c = 2 \). Therefore, using the rule, \[ \int 2 \, dx = 2x + C. \]

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!

Key Concepts

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

integration of constants
Integrating a constant is one of the simplest tasks in calculus. In this exercise, the constant is 2. When you're given the integral of a constant, it is the same as finding the area under a horizontal line on a graph.
The function doesn't change with the variable, so it's simply a flat line.
This is why we use a specific rule for integrating constants.
basic integration rules
Let's focus on the basic integration rule for constants. The rule states that if you have an integral of the form \[ \int c \, dx = cx + C \] where \( c \) is a constant.
It means you multiply the constant by the variable of integration, then add a constant of integration, represented as \( C \).
For example, in our problem, \( c = 2 \). Applying the rule, we get:

\[ \int 2 \, dx = 2x + C \]
constant of integration
When dealing with indefinite integrals, always remember to add the 'constant of integration'. This constant, denoted as 'C', represents the fact that there are infinitely many antiderivatives for any given function.
Each solution differs by a constant value. For instance, both expressions \[ f(x) = 2x + 3 \] and \[ f(x) = 2x + 4 \] are antiderivatives of \[ 2 \].
The 'C' accounts for these variations, ensuring your answer includes all possible solutions. Neglecting 'C' can lead to incomplete or incorrect solutions.
So, whenever you integrate, make it a habit to include the '+ C'.

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

Find \(s(t)\) $$a(t)=-2 t+6, \text { with } v(0)=6 \text { and } s(0)=10$$

Memorizing. The rate of memorizing information initially increases. Eventually, however, a maximum rate is reached, after which it begins to decrease. GRAPH CANT COPY Suppose that in a memory experiment the rate of memorizing is given by \(M^{\prime}(t)=-0.009 t^{2}+0.2 t\) where \(M^{\prime}(t)\) is the memory rate, in words per minute. How many words are memorized in the first 10 min (from \(t=0\) to \(t=10\) )?

Memorizing. The rate of memorizing information initially increases. Eventually, however, a maximum rate is reached, after which it begins to decrease. GRAPH CANT COPY Suppose that in another memory experiment the rate of memorizing is given by \(M^{\prime}(t)=-0.003 t^{2}+0.2 t\) where \(M^{\prime}(t)\) is the memory rate, in words per minute. How many words are memorized in the first 10 min (from \(t=0\) to \(t=10\) )?

Evaluate. Assume \(u>0\) when ln u appears. $$\int \frac{d x}{x(\ln x)^{4}}$$

Evaluate. \(\int_{0}^{2} \sqrt{2 x} d x \quad(\text { Hint: Simplify first. })\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.