Question

considere la función (f(x) = 4x^3 - 12x^2 + 6x - 5).
una antiderivada de (f(x)) es (f(x) = ax^4 + bx^3 + cx^2 + dx)

donde (a) es

y (b) es

y (c) es

y (d) es

Explanation:

Understand the given function

We are given the function:

$$f(x) = 4x^3 - 12x^2 + 6x - 5$$

We need to find an antiderivative \(F(x)\) of the form:

$$F(x) = Ax^4 + Bx^3 + Cx^2 + Dx$$

where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Apply the power rule for integration

To find the antiderivative of a polynomial, we apply the Power Rule for Integration term-by-term. The rule states that for any real number \(n
eq -1\):

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

Let's integrate each term of \(f(x)\):

1. For \(4x^3\):

$$\int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4$$

1. For \(-12x^2\):

$$\int -12x^2 \, dx = -12 \cdot \frac{x^3}{3} = -4x^3$$

1. For \(6x\):

$$\int 6x \, dx = 6 \cdot \frac{x^2}{2} = 3x^2$$

1. For \(-5\):

$$\int -5 \, dx = -5x$$

Combine terms and identify coefficients

Combining the integrated terms, we get the general antiderivative:

$$F(x) = x^4 - 4x^3 + 3x^2 - 5x$$

Comparing this with the given form:

$$F(x) = Ax^4 + Bx^3 + Cx^2 + Dx$$

We can directly identify the coefficients:

• \(A = 1\)
• \(B = -4\)
• \(C = 3\)
• \(D = -5\)

Answer:

Consider the function \(f(x) = 4x^3 - 12x^2 + 6x - 5\).
An antiderivative of \(f(x)\) is \(F(x) = Ax^4 + Bx^3 + Cx^2 + Dx\)

where \(A\) is <blank>1</blank>

and \(B\) is <blank>-4</blank>

and \(C\) is <blank>3</blank>

and \(D\) is <blank>-5</blank>