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Chapter 4: Problem 4

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=3 x^{2}+\sqrt{3} $$

Short Answer

Expert verified
The general antiderivative is \( F(x) = x^3 + \sqrt{3}x + C \).

Step by step solution

01

Identify the terms of the function

The given function is \( f(x) = 3x^2 + \sqrt{3} \). This function is composed of two terms: \( 3x^2 \) and \( \sqrt{3} \).
02

Apply the power rule to the first term

For the term \( 3x^2 \), we use the power rule of integration which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Therefore, the antiderivative of \( 3x^2 \) is \( \frac{3x^{2+1}}{2+1} = x^3 \).
03

Integrate the constant term

The antiderivative of a constant \( a \) is \( ax + C \). Therefore, the antiderivative of \( \sqrt{3} \) is \( \sqrt{3}x \).
04

Combine the results

Combine the antiderivatives from Steps 2 and 3 to form the general antiderivative. The combined antiderivative is \( F(x) = x^3 + \sqrt{3}x + C \).

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Key Concepts

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

Integration
Integration is a fundamental concept in calculus often introduced along with differentiation. In essence, integration is the reverse process of differentiation. It allows us to find a function given its derivative. The function obtained from integrating is called the antiderivative or integral.
  • The integral of a function gives the accumulation of the quantities, which can be thought of as finding the area under a curve represented by the function.
  • Definite integrals have limits of integration and compute a real value, while indefinite integrals do not have limits and provide a family of functions denoted by adding the constant of integration.
In our example, the task is to find the general antiderivative of the function given as a sum of terms. Different terms in a function can often be separately integrated and then summed. The process leverages formulas and rules developed for different types of functions, such as polynomials and constants, to simplify the integration process.
Power Rule
The Power Rule is a very useful tool in both differentiation and integration. In integration, it helps find antiderivatives of power functions, which are functions of the form \( x^n \), where \( n \) is any real number.
  • The Power Rule for integration states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where } n eq -1. \]
  • This means we simply add 1 to the power and then divide by the new power.
In the given exercise, this rule was applied to the term \( 3x^2 \) by taking \( n = 2 \). The antiderivative becomes \( \frac{3x^3}{3} = x^3 \). The rule simplifies the process, making it easy to handle polynomial functions.
Constant of Integration
In calculus, when we find an indefinite integral, we usually end up with a general solution that includes a constant term, known as the Constant of Integration. This is because the derivative of a constant is zero, which creates an infinite set of solutions that differ by a constant.
  • The Constant of Integration is typically denoted by \( C \) and represents any real number.
  • It ensures that the set of all possible original functions is captured, since adding any constant to a function doesn't change its derivative.
In our example, after integrating all terms of the function \( f(x) = 3x^2 + \sqrt{3} \), a \( C \) is added to form the final general antiderivative: \( F(x) = x^3 + \sqrt{3}x + C \). This expression includes the entire family of functions that could have \( f(x) \) as their derivative.
Mathematics Education
In mathematics education, understanding the process of integration and its rules is essential for students. Learning integration fosters a deeper comprehension of calculus and its applications.
  • Students build upon foundational math skills such as algebra and differentiation when learning to integrate.
  • Recognizing patterns and applying rules such as the Power Rule facilitate efficient problem solving.
  • Guided practice with examples, like the one provided here, aids students in mastering the topic by seeing step-by-step procedures and explanations.
The integration problem described here highlights practical applications of theory taught in mathematics education. It shows how abstract concepts are used in real problem-solving scenarios, enhancing the learning experience and preparing students for advanced studies.

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Most popular questions from this chapter

GG 44. The Census Bureau estimates that the growth rate \(k\) of the world population will decrease by roughly \(0.0002\) per year for the next few decades. In \(2004, k\) was \(0.0132\). (a) Express \(k\) as a function of time \(t\), where \(t\) is measured in years since 2004 . (b) Find a differential equation that models the population \(y\) for this problem. (c) Solve the differential equation with the additional condition that the population in \(2004(t=0)\) was \(6.4\) billion. (d) Graph the population \(y\) for the next 300 years. (e) With this model, when will the population reach a maximum? When will the population drop below the 2004 level?

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=\frac{x^{4}+1}{x^{3}} $$

Using the same axes, draw the graphs for \(0 \leq t \leq 100\) of the following two models for the growth of world population (both described in this section). (a) Exponential growth: \(y=6.4 e^{0.0132 t}\) (b) Logistic growth: \(y=102.4 /\left(6+10 e^{-0.030 t}\right)\) Compare what the two models predict for world population in 2010,2040, and 2090 . Note: Both models assume that world population was \(6.4\) billion in \(2004(t=0)\).

An object is moving along a coordinate line subject to the indicated acceleration a (in centimeters per second per second) with the initial velocity \(v_{0}\) (in centimeters per second) and directed distance \(s_{0}\) (in centimeters). Find both the velocity \(\underline{v}\) and directed distance \(s\) after 2 seconds (see Example 4). $$ a=\sqrt[3]{2 t+1} ; v_{0}=0, s_{0}=10 $$

Let \(E\) be a differentiable function satisfying \(E(u+v)=E(u) E(v)\) for all \(u\) and \(v .\) Find a formula for \(E(x) .\) Hint: First find \(E^{\prime}(x) .\)
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