Finding Antiderivative F(x) Of F(x) = X^6 - 3x^{-4} + 1 With F(1) = 2
In calculus, the antiderivative, also known as the indefinite integral, is the reverse process of differentiation. Given a function f(x), its antiderivative F(x) is a function whose derivative is f(x), i.e., F'(x) = f(x). The process of finding antiderivatives is fundamental in solving various problems in mathematics, physics, and engineering. This article delves into the step-by-step process of finding the antiderivative F(x) of the function f(x) = x^6 - 3x^{-4} + 1, subject to the condition F(1) = 2. We will explore the power rule of integration, the constant of integration, and how to determine its value using the given initial condition. Understanding these concepts is crucial for mastering integral calculus and its applications.
Understanding Antiderivatives and Indefinite Integrals
Before diving into the solution, it's crucial to understand the concept of antiderivatives and indefinite integrals. The antiderivative of a function f(x) is another function F(x) whose derivative is f(x). Mathematically, this is represented as F'(x) = f(x). The term indefinite integral is often used interchangeably with antiderivative, and it represents the family of all functions that have the same derivative. This family of functions differs only by a constant, known as the constant of integration, usually denoted as C.
When we find the antiderivative of a function, we are essentially reversing the process of differentiation. For instance, if the derivative of x^2 is 2x, then the antiderivative of 2x is x^2 + C, where C represents the constant of integration. This constant arises because the derivative of any constant is zero, so when we reverse the process, we need to account for this arbitrary constant. The power rule is a fundamental concept in finding antiderivatives, stating that the antiderivative of x^n (where n ≠-1) is (x^(n+1))/(n+1) + C. This rule is derived from the reverse of the power rule in differentiation, where the power is reduced by one. Understanding this rule is essential for integrating polynomial functions and other expressions involving powers of x.
In practical terms, antiderivatives are used extensively in solving differential equations, calculating areas under curves (definite integrals), and modeling various physical phenomena. For example, in physics, if we know the velocity function of an object, we can find its position function by taking the antiderivative of the velocity. Similarly, in economics, antiderivatives can be used to determine total cost or revenue functions from marginal cost or revenue functions. The constant of integration plays a critical role in these applications, as it allows us to specify a particular solution that satisfies a given initial condition. Initial conditions provide additional information, such as the value of the function at a specific point, which helps us determine the unique value of C. This makes antiderivatives not just theoretical constructs but powerful tools for solving real-world problems.
Step-by-Step Solution to Finding F(x)
To find the antiderivative F(x) of the given function f(x) = x^6 - 3x^{-4} + 1 that satisfies F(1) = 2, we will follow a step-by-step approach. This involves applying the power rule for integration to each term of the function, determining the constant of integration, and finally, using the given condition F(1) = 2 to find the specific value of the constant.
1. Applying the Power Rule of Integration
The first step in finding F(x) is to apply the power rule of integration to each term in f(x). The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, where n ≠-1 and C is the constant of integration. We apply this rule term by term:
- For the term x^6: Applying the power rule, we increase the exponent by 1 and divide by the new exponent. So, the antiderivative of x^6 is (x^(6+1))/(6+1) = x^7/7. This is a straightforward application of the power rule, where we increase the power from 6 to 7 and divide by the new power.
- **For the term -3x^-4}*** is (-3x^(-4+1))/(-4+1) = (-3x^(-3))/(-3) = x^(-3). Here, the negative exponent indicates that the term can also be written as 1/x^3.
- For the term 1: The antiderivative of a constant is simply the constant times x. So, the antiderivative of 1 is 1x = x. This is because the derivative of x is 1, so the reverse process gives us x.
By applying the power rule to each term, we find the antiderivative of each part of the function. This is a crucial step in the integration process, as it breaks down a complex function into simpler components that can be integrated individually. The constant of integration, C, is added at the end to represent the family of all possible antiderivatives.
2. Combining the Antiderivatives and Adding the Constant of Integration
Now that we have found the antiderivative of each term in f(x), we combine them and add the constant of integration C. This gives us the general form of the antiderivative F(x):
F(x) = x^7/7 + x^{-3} + x + C
This expression represents a family of functions, each differing by a constant value C. The constant C accounts for the fact that the derivative of a constant is always zero. Therefore, when finding an antiderivative, we need to include C to represent all possible constant terms that could have been present in the original function before differentiation.
This general form of the antiderivative is often sufficient for many applications, particularly when we are interested in properties that do not depend on the specific value of C. However, in situations where we have additional information, such as an initial condition, we can determine the specific value of C and find a unique antiderivative. This is particularly important in solving initial value problems in differential equations, where we need to find a specific solution that satisfies a given condition at a particular point.
3. Using the Condition F(1) = 2 to Find the Constant C
To find the specific antiderivative that satisfies the condition F(1) = 2, we substitute x = 1 into the general form of F(x) and set the result equal to 2. This allows us to solve for the constant of integration C:
F(1) = (1^7)/7 + (1^{-3}) + 1 + C = 2
Simplifying the equation:
1/7 + 1 + 1 + C = 2
Combining the terms:
1/7 + 2 + C = 2
Subtracting 2 from both sides:
1/7 + C = 0
Solving for C:
C = -1/7
This value of C is the specific constant that ensures our antiderivative F(x) satisfies the given condition F(1) = 2. By determining C, we move from the general antiderivative to a specific antiderivative, which is crucial for applications where initial conditions are known. This process of using initial conditions to find a specific solution is a common technique in calculus and differential equations.
4. Writing the Final Antiderivative F(x)
Now that we have found the constant of integration C = -1/7, we can write the final antiderivative F(x) by substituting this value back into the general form:
F(x) = x^7/7 + x^{-3} + x - 1/7
This is the specific antiderivative of f(x) = x^6 - 3x^{-4} + 1 that satisfies the condition F(1) = 2. This final expression represents a unique function that not only has f(x) as its derivative but also passes through the point (1, 2). This demonstrates the power of using initial conditions to pinpoint a particular solution from a family of possible antiderivatives.
In summary, finding the antiderivative involves applying integration rules, adding the constant of integration, and using given conditions to solve for the constant. This process is fundamental in calculus and is used extensively in various fields to solve problems involving rates of change and accumulation.
Common Mistakes to Avoid When Finding Antiderivatives
Finding antiderivatives can be tricky, and several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct answer. One of the most frequent errors is forgetting to add the constant of integration, C. As discussed earlier, the antiderivative is a family of functions that differ by a constant, so C is a crucial part of the solution. Omitting C means you are only finding one particular antiderivative, not the general form.
Another common mistake is misapplying the power rule for integration. The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, but this rule does not apply when n = -1. The antiderivative of x^{-1} (which is 1/x) is ln|x| + C, not x^0/0, which is undefined. Recognizing when to use the power rule and when to apply other integration techniques is essential. Incorrectly applying the power rule can lead to significant errors in your calculations.
Another source of errors is with negative exponents and algebraic manipulations. It's easy to make mistakes when dealing with negative exponents, especially when combined with fractions. Careful attention to detail and double-checking your algebra can help prevent these errors. Additionally, be cautious when simplifying expressions. Incorrect simplification can lead to a wrong final answer. Taking small, careful steps and verifying each step can help avoid these issues.
When dealing with initial conditions, a common mistake is substituting the value incorrectly or solving for C improperly. Make sure you substitute the given value of x into the antiderivative F(x) correctly and set the result equal to the given value of F(x). Then, solve the resulting equation carefully for C. A small error in this step can lead to an incorrect constant of integration and, consequently, a wrong specific antiderivative.
Finally, rushing through the problem without checking your work is a common mistake. After finding the antiderivative, take a moment to differentiate your answer to see if you get back the original function f(x). This is a simple way to verify that your integration is correct. If your derivative does not match the original function, you know there is an error somewhere in your work, and you can go back and review each step.
Real-World Applications of Antiderivatives
Antiderivatives are not just theoretical constructs; they have numerous real-world applications across various fields. One of the most prominent applications is in physics, where antiderivatives are used to relate velocity, acceleration, and position. If you know the acceleration of an object as a function of time, you can find its velocity by taking the antiderivative of the acceleration function. Similarly, you can find the position function by taking the antiderivative of the velocity function. These relationships are fundamental in understanding motion and are used in mechanics, kinematics, and other areas of physics.
In engineering, antiderivatives are used in a wide range of applications, including circuit analysis, signal processing, and control systems. For instance, in electrical engineering, if you know the current flowing through a capacitor, you can find the voltage across the capacitor by integrating the current with respect to time. In mechanical engineering, antiderivatives are used to calculate work done by a force, potential energy, and other quantities. These calculations are essential for designing and analyzing various engineering systems.
Economics also benefits from the use of antiderivatives. For example, if you know the marginal cost function of producing a certain item, you can find the total cost function by taking the antiderivative of the marginal cost function. Similarly, if you know the marginal revenue function, you can find the total revenue function. These applications help economists understand cost structures, revenue generation, and profit maximization.
In statistics and probability, antiderivatives are used to find cumulative distribution functions (CDFs) from probability density functions (PDFs). The CDF gives the probability that a random variable will take on a value less than or equal to a given value, and it is found by integrating the PDF. This is crucial for understanding the distribution of data and making probabilistic predictions.
Antiderivatives also play a significant role in computer graphics and image processing. For example, they can be used to perform image reconstruction, where an image is created from its gradient information. This is used in various applications, such as medical imaging and computer vision.
These examples illustrate the broad applicability of antiderivatives in solving real-world problems. The ability to reverse the process of differentiation allows us to move from rates of change to accumulated quantities, making antiderivatives a powerful tool in many disciplines.
Conclusion
In conclusion, finding the antiderivative F(x) of the function f(x) = x^6 - 3x^{-4} + 1 that satisfies the condition F(1) = 2 involves applying the power rule of integration, combining the antiderivatives, adding the constant of integration, and using the given condition to solve for the constant. By following these steps carefully, we found that F(x) = x^7/7 + x^{-3} + x - 1/7. This process demonstrates the fundamental principles of integral calculus and the importance of antiderivatives in solving various mathematical and real-world problems. Understanding and mastering these techniques is essential for anyone studying calculus and its applications in science, engineering, economics, and other fields.
