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Chapter 5: Problem 3

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

Short Answer

Expert verified
The general antiderivative of \(f(x) = x^2 + 3x - 4\) is \(F(x) = \frac{x^3}{3} + \frac{3x^2}{2} - 4x + C\).

Step by step solution

01

Recall the Antiderivative Formula

To find the antiderivative of a polynomial function, such as \[f(x) = x^2 + 3x - 4\]you need to apply the power rule for integration. The general rule for the antiderivative of x^n is \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where C is the constant of integration.
02

Integrate Each Term Separately

Break the function into its components and integrate each term individually:1. The antiderivative of x^2 is \[\frac{x^{2+1}}{2+1} = \frac{x^3}{3}\]2. The antiderivative of 3x is \[3 \cdot \frac{x^{1+1}}{1+1} = \frac{3x^2}{2}\]3. The antiderivative of -4 is \[-4x\](because the integral of a constant a is ax).
03

Combine the Results

Combine the antiderivatives of each term:\[\int (x^2 + 3x - 4) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} - 4x + C\]where C is the constant of integration.

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Key Concepts

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

Understanding the Power Rule for Integration
The power rule for integration is an essential building block in calculus. It helps to find antiderivatives of terms where the variable is raised to a power. This rule is handy for handling polynomial functions. The power rule states that if you need to integrate a function of the form \(x^n\), then the antiderivative is \(\frac{x^{n+1}}{n+1} + C\). Here, \(n eq -1\) because dividing by zero is undefined.
Key Points to Remember:
  • Increase the exponent by one for each term.
  • Divide by the new exponent.
  • Always add the constant of integration \(C\).
Using the power rule accurately simplifies the integration process, turning complex formulas into manageable calculations.
Exploring Polynomial Functions
Polynomial functions are made up of terms that are non-negative integer powers of variables. Commonly, these functions take the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). Here, each \(a\) represents a constant coefficient, while \(x\) stands for the variable.
In the problem scenario, the function is \(f(x) = x^2 + 3x - 4\).
  • This function includes a quadratic term \(x^2\), a linear term \(3x\), and a constant term \(-4\).
  • Polynomial functions are straightforward to integrate using the power rule because each term can be handled separately.
Understand the structure of polynomial functions as it helps in breaking down complex expressions into simpler, integrable parts.
The Role of the Constant of Integration
When we integrate a function, we always append a constant, denoted as \(C\), to the result. This is because the process of integration, unlike differentiation, is not unique. Any constant can be added to a function without changing its derivative. Thus, the general solution to an indefinite integral includes this constant of integration.
  • Without the constant \(C\), the solution to an integral would only represent one possible antiderivative.
  • The presence of \(C\) ensures that we account for all possible vertical shifts of the original function.
Appreciating the role of \(C\) is crucial for understanding the complete set of solutions that an indefinite integral can provide.

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