Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.


f(x) = 8x³ + sin x; F(0) = 2

Verified step by step guidance

1. Identify the given function to integrate: \(f(x) = 8x^{3} + \sin x\).
2. Find the general antiderivative \(F(x)\) by integrating each term separately: \(\int 8x^{3} \, dx\) and \(\int \sin x \, dx\).
3. Recall the integral formulas: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\), and \(\int \sin x \, dx = -\cos x + C\).
4. Apply these formulas to get \(F(x) = 2x^{4} - \cos x + C\), where \(C\) is the constant of integration.
5. Use the initial condition \(F(0) = 2\) to solve for \(C\) by substituting \(x=0\) into \(F(x)\) and setting the expression equal to 2.

Key Concepts

Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). It represents the reverse process of differentiation and is expressed as an indefinite integral, including an arbitrary constant C since differentiation eliminates constants.

Initial Condition for Particular Solution

An initial condition like F(0) = 2 specifies the value of the antiderivative at a particular point, allowing us to determine the constant of integration C. This transforms the general antiderivative into a unique particular solution.

Integration of Basic Functions

To find the antiderivative, one must integrate each term separately using known formulas: the integral of x^n is (x^(n+1))/(n+1), and the integral of sin x is -cos x. Combining these results forms the general antiderivative.