Find the Antiderivative f(y)=9+12y+4y^2
The problem is asking for the determination of the antiderivative (also known as the indefinite integral) of the given function f(y) with respect to the variable y. The function f(y) is a polynomial, expressed as 9 + 12y + 4y^2. The task is to find a function F(y) such that F'(y) = f(y), meaning the derivative of F(y) with respect to y is equal to the given function f(y). The solution will involve applying the basic rules of antiderivatives for each term of the polynomial.
$f \left(\right. y \left.\right) = 9 + 12 y + 4 y^{2}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(y)$ by integrating the given function $f(y)$.
$$F(y) = \int f(y) \, dy$$
Step 2:
Construct the integral for the given function.
$$F(y) = \int (9 + 12y + 4y^2) \, dy$$
Step 3:
Decompose the integral into separate terms.
$$\int 9 \, dy + \int 12y \, dy + \int 4y^2 \, dy$$
Step 4:
Apply the integration rule for constants.
$$9y + C + \int 12y \, dy + \int 4y^2 \, dy$$
Step 5:
Factor out the constant $12$ from the integral.
$$9y + C + 12\int y \, dy + \int 4y^2 \, dy$$
Step 6:
Utilize the Power Rule for integration on $y$.
$$9y + C + 12\left(\frac{y^2}{2} + C\right) + \int 4y^2 \, dy$$
Step 7:
Factor out the constant $4$ from the integral.
$$9y + C + 12\left(\frac{y^2}{2} + C\right) + 4\int y^2 \, dy$$
Step 8:
Use the Power Rule for integration on $y^2$.
$$9y + C + 12\left(\frac{y^2}{2} + C\right) + 4\left(\frac{y^3}{3} + C\right)$$
Step 9:
Proceed to simplify the expression.
Step 9.1:
Combine terms.
$$9y + 6y^2 + 4\left(\frac{y^3}{3}\right) + C$$
Step 9.2:
Further simplify the expression.
Step 9.2.1:
Merge the fraction with the cubic term.
$$9y + 6y^2 + \frac{4y^3}{3} + C$$
Step 9.2.2:
Simplify the coefficient with the cubic term.
$$9y + 6y^2 + \frac{4y^3}{3} + C$$
Step 9.3:
Arrange the terms in ascending order of power.
$$9y + 6y^2 + \frac{4}{3}y^3 + C$$
Step 10:
Conclude with the antiderivative of the function $f(y) = 9 + 12y + 4y^2$.
$$F(y) = 9y + 6y^2 + \frac{4}{3}y^3 + C$$
Solution:"The antiderivative of the function $f(y) = 9 + 12y + 4y^2$ is $F(y) = 9y + 6y^2 + \frac{4}{3}y^3 + C$."
Knowledge Notes:
The antiderivative, also known as the indefinite integral, is the reverse process of differentiation.
The constant of integration $C$ is added because the antiderivative is not unique; any constant can be added to a function without changing its derivative.
The Power Rule for integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$.
Constants can be factored out of the integral, which simplifies the integration process.
When integrating a polynomial term by term, the result is a polynomial where each term's exponent has been increased by one, and the coefficient has been divided by the new exponent.
After integrating, it's important to simplify and combine like terms to get the final antiderivative in its simplest form.
