Finding The Antiderivative Of F(x) = 10x^8 + 7x^7 - 8x^4 - 2

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In the realm of calculus, the concept of an antiderivative holds paramount importance. It's the reverse operation of differentiation, allowing us to find a function whose derivative is a given function. In this exploration, we delve into the intricacies of finding the antiderivative of the polynomial function f(x) = 10x^8 + 7x^7 - 8x^4 - 2. Our objective is to determine the antiderivative, denoted as F(x), and express it in the form F(x) = Ax^n + Bx^m + Cx^p + Dx^q, where A, B, C, D, n, m, p, and q are constants to be determined. We'll set the constant of integration, C, to 0 for this specific case.

Before we embark on the journey of finding the antiderivative of our given function, let's solidify our understanding of what an antiderivative truly represents. In essence, an antiderivative is a function whose derivative is equal to the original function. Think of it as undoing the process of differentiation. For instance, if the derivative of a function is 2x, then an antiderivative of 2x would be x^2, because the derivative of x^2 is indeed 2x. However, it's crucial to acknowledge that antiderivatives are not unique. The function x^2 + 1 would also be an antiderivative of 2x, as its derivative is also 2x. This is because the derivative of a constant is always zero. Therefore, when finding an antiderivative, we typically add a constant of integration, often denoted as 'C', to account for this ambiguity.

The power rule of integration serves as a cornerstone in our quest to find the antiderivative of polynomial functions. This rule provides a direct method for integrating terms of the form x^n, where n is any real number except -1. The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule stems directly from the power rule of differentiation, which states that the derivative of x^n is nx^(n-1). The power rule of integration essentially reverses this process. To illustrate, let's consider the term x^3. Applying the power rule, we add 1 to the exponent (3 + 1 = 4) and divide by the new exponent (4), resulting in (x^4)/4. This signifies that (x^4)/4 is an antiderivative of x^3. The power rule provides a systematic approach to integrating polynomial terms, simplifying the process of finding antiderivatives.

Now, let's put our understanding of antiderivatives and the power rule into action by finding the antiderivative of our given function, f(x) = 10x^8 + 7x^7 - 8x^4 - 2. We'll apply the power rule to each term individually, keeping in mind the constant of integration. For the term 10x^8, we add 1 to the exponent (8 + 1 = 9) and divide by the new exponent (9), yielding (10x^9)/9. Similarly, for the term 7x^7, we add 1 to the exponent (7 + 1 = 8) and divide by the new exponent (8), resulting in (7x^8)/8. For the term -8x^4, we add 1 to the exponent (4 + 1 = 5) and divide by the new exponent (5), giving us (-8x^5)/5. Finally, for the constant term -2, we recognize that its antiderivative is -2x, as the derivative of -2x is -2. Combining these results, we obtain the antiderivative F(x) = (10x^9)/9 + (7x^8)/8 - (8x^5)/5 - 2x + C, where C is the constant of integration. For this specific case, we'll set C = 0.

Our next step involves expressing the antiderivative F(x) = (10x^9)/9 + (7x^8)/8 - (8x^5)/5 - 2x in the specified form F(x) = Ax^n + Bx^m + Cx^p + Dx^q. This involves identifying the coefficients and exponents corresponding to each term. By carefully comparing the two expressions, we can directly extract the values. The term (10x^9)/9 corresponds to Ax^n, where A = 10/9 and n = 9. The term (7x^8)/8 corresponds to Bx^m, where B = 7/8 and m = 8. The term (-8x^5)/5 corresponds to Cx^p, where C = -8/5 and p = 5. Lastly, the term -2x corresponds to Dx^q, where D = -2 and q = 1. We have successfully determined the values of A, B, C, D, n, m, p, and q, allowing us to express the antiderivative in the desired form.

  • A is 10/9
  • n is 9
  • B is 7/8

In this comprehensive exploration, we've successfully navigated the process of finding the antiderivative of the polynomial function f(x) = 10x^8 + 7x^7 - 8x^4 - 2. We've solidified our understanding of antiderivatives as the reverse operation of differentiation and mastered the application of the power rule of integration, a fundamental tool for integrating polynomial terms. By systematically applying the power rule to each term of the function, we derived the antiderivative F(x) = (10x^9)/9 + (7x^8)/8 - (8x^5)/5 - 2x. Furthermore, we expressed this antiderivative in the desired form F(x) = Ax^n + Bx^m + Cx^p + Dx^q, accurately identifying the coefficients and exponents for each term. This exercise highlights the significance of antiderivatives in calculus and underscores the importance of mastering integration techniques. The ability to find antiderivatives opens doors to solving a wide array of problems in mathematics, physics, engineering, and beyond. Understanding and applying these concepts is crucial for anyone venturing further into the world of calculus and its applications.