Question:

Find the antiderivative of the function f(x) = 3x^2 - 4x + 5.

Answer with Step-by-Step Explanation:

To find the antiderivative of the given function, we need to find the function F(x) such that F'(x) = f(x). In other words, we need to find the function F(x) whose derivative is equal to f(x).

The antiderivative of a term ax^n is given by (a/n+1) * x^(n+1) + C, where C is the constant of integration.

So, let's find the antiderivative of each term of the given function:

1. For the term 3x^2, the antiderivative is (3/3) * x^3 = x^3.
2. For the term -4x, the antiderivative is (-4/1) * x^1 = -4x.
3. For the constant term 5, the antiderivative is 5x.

Combining these results, the antiderivative of the function f(x) = 3x^2 - 4x + 5 is:

F(x) = x^3 - 4x + 5x + C, where C is the constant of integration.

Therefore, the antiderivative of the function f(x) = 3x^2 - 4x + 5 is F(x) = x^3 - 4x + 5x + C.