Finding The Antiderivative Of F(x) = X³ + X Given F(1) = 6 A Step-by-Step Solution

by Scholario Team 83 views

Understanding Antiderivatives

In the realm of calculus, the concept of an antiderivative stands as a cornerstone, representing the inverse operation of differentiation. At its heart, finding an antiderivative involves reversing the process of taking a derivative. Given a function f(x), its antiderivative, denoted as F(x), is a function whose derivative equals f(x). In simpler terms, F'(x) = f(x). This seemingly straightforward concept unlocks a vast array of applications, from solving differential equations to computing areas under curves. The antiderivative is not a single function but rather a family of functions differing by a constant. This constant, often represented as C, arises because the derivative of a constant is always zero. Consequently, when we reverse the process of differentiation, we introduce this constant to account for all possible functions that could have led to the given derivative. To truly grasp the significance of antiderivatives, it's essential to delve deeper into their properties and how they interact with other calculus concepts. The process of finding an antiderivative is known as integration, and it's a fundamental tool in solving various problems across mathematics, physics, engineering, and economics. Understanding antiderivatives also paves the way for exploring definite integrals, which provide a means to calculate the exact area under a curve between two specified points. This connection between antiderivatives and areas is a testament to the power and elegance of calculus. Moreover, the concept of antiderivatives extends beyond simple polynomial functions and encompasses a wide range of functions, including trigonometric, exponential, and logarithmic functions. Each type of function has its own set of rules and techniques for finding antiderivatives, making the study of integration a rich and rewarding endeavor. Ultimately, mastering antiderivatives is crucial for anyone seeking to deepen their understanding of calculus and its applications. It's a fundamental skill that empowers us to solve complex problems and gain insights into the world around us. Whether you're a student, a researcher, or simply someone curious about mathematics, the journey of exploring antiderivatives is sure to be both challenging and enlightening.

Problem Statement: Finding the Antiderivative

Our specific problem involves finding the antiderivative F(x) of the function f(x) = x³ + x, subject to the condition F(1) = 6. This condition provides a crucial piece of information that allows us to determine the specific antiderivative from the family of possible solutions. The function f(x) = x³ + x is a polynomial function, which makes it relatively straightforward to find its antiderivative using the power rule of integration. The power rule states that the antiderivative of xⁿ is (xⁿ⁺¹)/(n+1), where n is any real number except -1. Applying this rule to each term in f(x), we can find the general form of the antiderivative. However, the general form will include the constant of integration, C, which needs to be determined using the given condition F(1) = 6. This condition essentially acts as an initial value, allowing us to pinpoint the specific antiderivative that satisfies the problem's requirements. The process of finding the antiderivative and then using the initial condition is a common technique in calculus, particularly when solving differential equations. Differential equations are equations that involve derivatives of unknown functions, and finding the solutions often involves finding antiderivatives. In this case, the problem can be viewed as a simple differential equation: F'(x) = x³ + x, with the initial condition F(1) = 6. Solving this type of problem requires a solid understanding of antiderivatives, the power rule of integration, and how to use initial conditions to find specific solutions. It's a fundamental exercise that reinforces the core concepts of calculus and provides a foundation for tackling more complex problems. By carefully applying the power rule and utilizing the given condition, we can arrive at the unique antiderivative that satisfies the problem statement. This process not only demonstrates the power of calculus but also highlights the importance of precision and attention to detail in mathematical problem-solving. Ultimately, the solution to this problem serves as a valuable example of how antiderivatives are used to find functions that meet specific criteria, a skill that is essential in various fields of science and engineering.

Step-by-Step Solution

  1. Apply the Power Rule: To find the antiderivative F(x) of f(x) = x³ + x, we apply the power rule to each term. The power rule states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1 and C is the constant of integration. Applying this rule to x³, we get (x⁴)/4. Applying it to x (which is x¹), we get (x²)/2. Therefore, the antiderivative of f(x) is F(x) = (x⁴)/4 + (x²)/2 + C. This represents the general form of the antiderivative, which includes the constant of integration, C. The constant C arises because the derivative of any constant is zero, so there are infinitely many possible antiderivatives that differ only by a constant. To find the specific antiderivative that satisfies the given condition, we need to determine the value of C. This is where the initial condition F(1) = 6 comes into play. The power rule is a fundamental tool in integration, and it's essential to understand how to apply it correctly. It's also important to remember the constant of integration, as it's a crucial part of the general solution. Without the constant of integration, we would only have one possible antiderivative, which may not be the one that satisfies the given conditions. The next step involves using the initial condition to find the specific value of C, which will give us the unique antiderivative that solves the problem. This process demonstrates the importance of initial conditions in determining specific solutions to differential equations and other calculus problems. By carefully applying the power rule and keeping track of the constant of integration, we can successfully find the antiderivative of the given function.

  2. Use the Initial Condition: We are given that F(1) = 6. This means that when x = 1, the value of the antiderivative F(x) is 6. We can use this information to solve for the constant of integration, C. Substitute x = 1 into the general form of the antiderivative we found in the previous step: F(1) = (1⁴)/4 + (1²)/2 + C. Simplify this expression: F(1) = 1/4 + 1/2 + C. Since F(1) = 6, we can set up the equation: 6 = 1/4 + 1/2 + C. To solve for C, first find a common denominator for the fractions, which is 4. Rewrite the equation as: 6 = 1/4 + 2/4 + C. Combine the fractions: 6 = 3/4 + C. Now, subtract 3/4 from both sides of the equation to isolate C: C = 6 - 3/4. To subtract the fraction from the whole number, rewrite 6 as a fraction with a denominator of 4: 6 = 24/4. Now the equation becomes: C = 24/4 - 3/4. Subtract the fractions: C = 21/4. This is the value of the constant of integration that satisfies the given condition. By using the initial condition, we have successfully determined the specific antiderivative from the family of possible solutions. The initial condition provides a crucial piece of information that allows us to pinpoint the unique function that meets the problem's requirements. This step highlights the importance of initial conditions in calculus and differential equations. Without an initial condition, we would only have the general form of the antiderivative, which includes the constant of integration. With the initial condition, we can find the specific solution that passes through a given point or satisfies a particular value at a given point. Now that we have found the value of C, we can substitute it back into the general form of the antiderivative to obtain the final solution.

  3. Substitute C and Final Answer: Now that we have found the value of the constant of integration, C = 21/4, we can substitute it back into the general form of the antiderivative F(x) = (x⁴)/4 + (x²)/2 + C. Substituting C = 21/4, we get: F(x) = (x⁴)/4 + (x²)/2 + 21/4. This is the specific antiderivative of f(x) = x³ + x that satisfies the condition F(1) = 6. This function represents the solution to the problem, and it's the unique function whose derivative is f(x) and whose value at x = 1 is 6. The final answer is a polynomial function with a constant term, and it demonstrates how the initial condition plays a crucial role in determining the specific antiderivative. The process of finding the antiderivative and then using the initial condition is a fundamental technique in calculus, and it's used extensively in solving differential equations and other problems. This example provides a clear illustration of how to apply the power rule of integration and how to use initial conditions to find specific solutions. The final answer, F(x) = (x⁴)/4 + (x²)/2 + 21/4, is the unique function that satisfies the given conditions, and it represents the solution to the problem. This result highlights the power of calculus in finding functions that meet specific criteria, and it demonstrates the importance of precision and attention to detail in mathematical problem-solving. By carefully following the steps and applying the appropriate techniques, we have successfully found the antiderivative of the given function and verified that it satisfies the given condition.

Conclusion

In conclusion, the antiderivative of f(x) = x³ + x with F(1) = 6 is F(x) = (x⁴)/4 + (x²)/2 + 21/4. This solution was obtained by applying the power rule of integration to find the general form of the antiderivative, and then using the initial condition F(1) = 6 to determine the specific value of the constant of integration. The process involved substituting the value x = 1 into the general form of the antiderivative, setting the result equal to 6, and solving for the constant C. This allowed us to pinpoint the unique antiderivative that satisfies the given condition. The problem highlights the fundamental concepts of calculus, including antiderivatives, integration, and the power rule. It also demonstrates the importance of initial conditions in determining specific solutions to problems involving antiderivatives and differential equations. The antiderivative, also known as the indefinite integral, represents the reverse process of differentiation. Finding antiderivatives is a crucial skill in calculus, with applications in various fields such as physics, engineering, and economics. Understanding the concept of antiderivatives is essential for solving differential equations, calculating areas under curves, and modeling real-world phenomena. The constant of integration, C, plays a vital role in the general form of the antiderivative. It represents the family of functions that have the same derivative. The initial condition provides the necessary information to determine the specific value of C and obtain the unique antiderivative that satisfies the given requirements. By carefully applying the rules of integration and utilizing initial conditions, we can successfully solve a wide range of problems involving antiderivatives. This example serves as a valuable illustration of the process and reinforces the core concepts of calculus. The final solution, F(x) = (x⁴)/4 + (x²)/2 + 21/4, is a polynomial function that represents the antiderivative of f(x) = x³ + x and satisfies the condition F(1) = 6. This result demonstrates the power of calculus in finding functions that meet specific criteria and highlights the importance of precision and attention to detail in mathematical problem-solving.