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Chapter 5: Problem 7

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=4 x^{3}-2 x+3 $$

Short Answer

Expert verified
The general antiderivative is \( x^4 - x^2 + 3x + C \).

Step by step solution

01

Understanding the Problem

We need to find the general antiderivative of the function \( f(x)=4x^3-2x+3 \). The antiderivative of a function, also known as the indefinite integral, is a function whose derivative is the given function.
02

Applying the Power Rule for Antiderivatives

The power rule for antiderivatives states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), provided \( n eq -1 \). We will apply this rule to each term of \( f(x)=4x^3-2x+3 \).
03

Finding the Antiderivative of the First Term

The first term is \( 4x^3 \). According to the power rule, its antiderivative is \( \frac{4x^{4}}{4} = x^4 \).
04

Finding the Antiderivative of the Second Term

The second term is \( -2x \). Its antiderivative is \( \frac{-2x^{2}}{2} = -x^2 \).
05

Finding the Antiderivative of the Constant Term

The constant term is \( 3 \). The antiderivative of a constant \( c \) is \( cx \). Thus, the antiderivative is \( 3x \).
06

Combining the Antiderivative Terms

Add the antiderivatives from each term together and include the constant of integration \( C \), resulting in the general antiderivative: \( x^4 - x^2 + 3x + C \).

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Key Concepts

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

Indefinite Integrals
An indefinite integral is the antiderivative of a function, providing a family of functions that differ only by a constant. This constant is often represented by the symbol \( C \). In calculus, when we find the indefinite integral of a function like \( f(x) = 4x^3 - 2x + 3 \), we are looking for a function \( F(x) \) such that \( F'(x) = f(x) \). The indefinite integral is written using the integral sign: \( \int f(x) \, dx \).
  • When calculating indefinite integrals, every result will include the constant of integration, \( C \), since differentiating a constant gives zero.
  • Unlike definite integrals, indefinite integrals do not have upper and lower limits.
  • The process results in general expressions representing a wide class of functions sharing the same derivative.
Power Rule
The power rule is a fundamental technique in calculus used to find the antiderivative of power functions, which are functions of the form \( x^n \). It states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) as long as \( n eq -1 \). This rule makes it easier to work with polynomial terms.
Applying the power rule involves:
  • Increasing the exponent by 1.
  • Dividing by the new exponent.
  • Adding the constant of integration \( C \) to account for any constant term that was differentiated away.
For example:
  • For \( x^3 \), the antiderivative is \( \frac{x^{4}}{4} \), because we increase 3 to 4 and divide by 4.
  • For a constant \( c \), its antiderivative is \( cx \), because the derivative of \( cx \) is simply \( c \).
Calculus
Calculus is a branch of mathematics focused on the principles of change and motion, providing tools to understand and describe such phenomena. It is commonly divided into two main areas: differential calculus and integral calculus.
  • Differential Calculus: Involves finding derivatives, which are rates of change. It's used to determine the slope of a curve at any given point or to solve various problems involving instantaneous change.
  • Integral Calculus: Concerns finding antiderivatives and integrals. This process primarily involves determining areas under curves and accumulating quantities.
Antiderivatives are central to integral calculus. When you find an antiderivative, you're essentially solving the problem of reversing differentiation to find the original function. Indefinite integrals help mathematicians and engineers capture more general solutions that apply across different circumstances or initial conditions.

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Most popular questions from this chapter

Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0 $$ for any number \(p>0 .\) This shows that the logarithmic function grows more slowly than any positive power of \(x\) as \(x \rightarrow \infty\).

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