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Magazine Chapter 2: Problem 2
Find an integral \(F\) of \(f(x)=5 x^{4}-4 x\) which satisfies \(F(2)=\) 12
Short Answer
Expert verified
The integral is \( F(x) = x^5 - 2x^2 - 12 \).
Step by step solution
01
Find the Indefinite Integral
To find an antiderivative \( F(x) \) of the function \( f(x) = 5x^4 - 4x \), we integrate the function with respect to \( x \). The indefinite integral is \( \int (5x^4 - 4x) \, dx \). Using the power rule, integrate each term separately: \[ \int 5x^4 \, dx = \frac{5}{5}x^{5} = x^5 \] and \[ \int -4x \, dx = -4 \cdot \frac{x^2}{2} = -2x^2 \]. Therefore, the antiderivative is \( F(x) = x^5 - 2x^2 + C \), where \( C \) is the constant of integration.
02
Use the Initial Condition
We're given the condition \( F(2) = 12 \). Substitute \( x = 2 \) into the antiderivative equation to solve for \( C \): \[ F(2) = (2)^5 - 2(2)^2 + C = 12 \]. Simplifying yields \( 32 - 8 + C = 12 \), which simplifies further to \( 24 + C = 12 \). Solving for \( C \) gives \( C = 12 - 24 = -12 \).
03
Write the Specific Integral
Now that we have solved for \( C \), write the specific integral that satisfies the initial condition. Using the form \( F(x) = x^5 - 2x^2 + C \), substitute \( C = -12 \) to get: \[ F(x) = x^5 - 2x^2 - 12 \]. This is the integral that satisfies \( F(2) = 12 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indefinite Integral
An indefinite integral refers to the process of finding the antiderivative of a given function. When you see the integral symbol \( \int \), it's indicating that you need to find a function whose derivative matches the function inside the integral. In our exercise, the task is to determine an antiderivative for the function \( f(x) = 5x^4 - 4x \).
Steps to find an indefinite integral:
Steps to find an indefinite integral:
- Identify the function you want to integrate.
- Use integration rules, such as the power rule, to find the antiderivative.
- Don't forget to add a constant of integration \( C \) at the end of your result, because indefinite integrals can represent a family of functions.
Power Rule
The power rule is a straightforward method used in calculus to integrate terms of the form \( x^n \). It simplifies the process of integration, making it easy to determine the antiderivative of polynomial functions. Here鈥檚 how it works:
- When integrating \( x^n \), apply the rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \).
- This rule requires you to add 1 to the exponent and then divide by the new exponent.
Antiderivative
An antiderivative of a function is another function whose derivative is the original function. Finding an antiderivative is key in solving integrals. Consider our exercise function \( f(x) = 5x^4 - 4x \). After integration, we establish that the antiderivative is \( F(x) = x^5 - 2x^2 + C \).
Important points about antiderivatives:
Important points about antiderivatives:
- They represent the most general form of the solution to an indefinite integral.
- Every continuous function has at least one antiderivative.
- The antiderivative is not unique unless an initial condition is given.
Initial Condition
Initial conditions help you determine the specific solution from a general family of solutions. They are values given for the function at a particular point, allowing you to solve for the constant \( C \) when finding an antiderivative.
In this problem, we are told that \( F(2) = 12 \). Using this condition:
In this problem, we are told that \( F(2) = 12 \). Using this condition:
- Substitute \( x = 2 \) into the antiderivative expression \( F(x) = x^5 - 2x^2 + C \).
- Set the equation equal to 12, which gives \( 32 - 8 + C = 12 \).
- Solve for \( C \), resulting in \( C = -12 \).
