91Ó°ÊÓ

In calculus, finding the antiderivative, also known as integration, is the reverse operation of differentiation. When we have a derivative like:\[ f^{\text{'} }(x) = 8x^{2} + 4x - 2 \]we look for a function, say \( f(x) \), whose derivative is the given function. To do this, we integrate the function. The indefinite integral of a function involves finding the original function.First, we write the integral:\[ f(x) = \int (8 x^{2}+4 x-2) \, dx \]Integrating term by term, we get:\[ f(x) = \frac{8}{3}x^3 + 2x^2 - 2x + C \]Here, \( C \) is the constant of integration. This constant is important and unavoidable in indefinite integrals. The next step involves determining this constant.

Initial conditions are values given at particular points to find specific solutions for differential equations and integrals. In our problem, the initial condition provided is \( f(0) = 6 \). This means that when \( x \) is zero, \( f(x) \) equals 6.To use this initial condition, we substitute \( x = 0 \) into our integrated function:\[ f(x) = \frac{8}{3}x^3 + 2x^2 - 2x + C \]So at \( x = 0 \):\[ f(0) = \frac{8}{3}(0)^3 + 2(0)^2 - 2(0) + C = 6 \]This simplifies to \( C = 6 \). This condition helps us find the particular solution to our problem by giving us the exact value for the constant of integration.

The constant of integration, denoted as \( C \), represents an infinite set of possible constants that can be added to an antiderivative. When integrating, you get a family of functions, each differing by a constant value because the derivative of any constant is zero.In our example, by applying the initial condition \( f(0) = 6 \), we found the exact value for \( C \), which is 6. Now we can write the particular solution:\[ f(x) = \frac{8}{3}x^3 + 2x^2 - 2x + 6 \]Without the initial condition, \( C \) could be any real number. But with \( f(0) = 6 \), \( C \) is uniquely determined. This is an essential step in solving many calculus problems, ensuring the solution is specific and accurate.