91Ó°ÊÓ

Calculus is a branch of mathematics that deals with rates of change and accumulation. It consists of two primary operations: differentiation and integration. Differentiation determines the rate at which a quantity changes, while integration focuses on finding the quantity when the rate of change is known. In this exercise, we're dealing with integration, which is basically the inverse operation of differentiation.

The problem begins with a derivative function, which tells us how the original function changes at any given point. By integrating this derivative, we "reverse" the process of differentiation. This allows us to find the original function, known as the antiderivative or integral, to which we only add a constant because integration is indefinite. The constant, labeled as \(C\), accounts for the vertical shift in all potential antiderivatives of the function. Each function derived from an antiderivative can differ by this constant value.

Integration is a fundamental concept in calculus that involves combining small quantities to find totals, like areas under curves or the original function from a given rate of change (derivative). Integrating turns a rate formula back into an accumulation formula or function, often called the antiderivative.

In the given problem, the task is to integrate the function \( f'(x) = 8x^3 + 12x + 3 \). This involves finding an antiderivative of each term separately:

• The integration of \(8x^3\) is \(2x^4\).
• The integration of \(12x\) is \(6x^2\).
• The integration of \(3\) is \(3x\).

Combining these, the antiderivative is \(f(x) = 2x^4 + 6x^2 + 3x + C\). The constant \(C\) represents any fixed number added to the function. This constant accounts for all potential vertical shifts of the function that still meet the derivative criteria.

An initial condition is a specific piece of information given in a problem that enables us to find the exact value of the constant of integration \(C\). It tells us a particular point on the function's graph. In this exercise, the initial condition is \(f(1) = 6\). This means when \(x = 1\), the value of \(f(x)\) is 6.

To use the initial condition, substitute \(x = 1\) into the antiderivative found: \(f(1) = 2(1)^4 + 6(1)^2 + 3(1) + C\). Simplifying on the left side gives \(11 + C\). Setting this equal to the given \(f(1) = 6\) allows us to solve for \(C\).

Subtract \(11\) from \(6\) to find \(C = -5\). This specific value of \(C\) ensures that the function satisfies both the derivative requirements and the initial condition. Therefore, the final function is \(f(x) = 2x^4 + 6x^2 + 3x - 5\). This includes every detail given in the exercise.