Find the Antiderivative f(x)=x(9-x)^2

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the function $f(x)$.
$F(x) = \int f(x) \, dx$

Step 2:

Write down the integral that needs to be solved.
$F(x) = \int x(9 - x)^2 \, dx$

Step 3:

Perform a substitution to simplify the integral. Let $u = 9 - x$, then $du = -dx$ which implies $-du = dx$.

Step 3.1:

Define the substitution $u = 9 - x$ and calculate $\frac{du}{dx}$.

Step 3.1.1:

Take the derivative of $9 - x$.
$\frac{d}{dx}(9 - x)$

Step 3.1.2:

Differentiate the expression.

Step 3.1.2.1:

Apply the Sum Rule to find the derivative of $9 - x$ with respect to $x$.
$\frac{d}{dx}(9) + \frac{d}{dx}(-x)$

Step 3.1.2.2:

The derivative of a constant is zero, so the derivative of $9$ with respect to $x$ is $0$.
$0 + \frac{d}{dx}(-x)$

Step 3.1.3:

Compute $\frac{d}{dx}(-x)$.

Step 3.1.3.1:

The derivative of $-x$ with respect to $x$ is $-1$ times the derivative of $x$ with respect to $x$.
$0 - \frac{d}{dx}(x)$

Step 3.1.3.2:

Apply the Power Rule which states that the derivative of $x^n$ is $nx^{n-1}$ where $n = 1$.
$0 - 1 \cdot 1$

Step 3.1.3.3:

Multiply $-1$ by $1$.
$0 - 1$

Step 3.1.4:

Combine the terms to get $-1$.
$-1$

Step 3.2:

Express the integral in terms of $u$ and $du$.
$\int -(-u + 9)u^2 \, du$

Step 4:

Expand the expression $-(-u + 9)u^2$.

Step 4.1:

Apply the distributive property to the expression.
$\int (u - 9)u^2 \, du$

Step 4.2:

Distribute the negative sign.
$\int u^3 - 9u^2 \, du$

Step 5:

Decompose the integral into simpler integrals.
$\int u^3 \, du - 9\int u^2 \, du$

Step 6:

Integrate $u^3$ with respect to $u$ using the Power Rule.
$\frac{1}{4}u^4 + C - 9\int u^2 \, du$

Step 7:

Take the constant $-9$ outside the integral.
$\frac{1}{4}u^4 + C - 9\int u^2 \, du$

Step 8:

Integrate $u^2$ with respect to $u$ using the Power Rule.
$\frac{1}{4}u^4 + C - 9\left(\frac{1}{3}u^3 + C\right)$

Step 9:

Simplify the expression.

Step 9.1:

Combine like terms.
$\frac{1}{4}u^4 - 3u^3 + C$

Step 10:

Substitute back $u = 9 - x$ into the antiderivative.
$\frac{1}{4}(9 - x)^4 - 3(9 - x)^3 + C$

Step 11:

Conclude with the antiderivative $F(x)$ of the function $f(x) = x(9 - x)^2$.
$F(x) = \frac{1}{4}(9 - x)^4 - 3(9 - x)^3 + C$

Knowledge Notes:

The problem involves finding the antiderivative of a given function, which is the reverse process of differentiation. The antiderivative is also known as the indefinite integral. The process of integration is used to find the antiderivative.

1. Integration: The process of finding the function $F(x)$ whose derivative is the given function $f(x)$.

2. Substitution Method: A technique used to simplify integrals by substituting a part of the integral with a new variable.

3. Sum Rule for Derivatives: The derivative of a sum of functions is the sum of the derivatives of those functions.

4. Power Rule for Derivatives: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

5. Power Rule for Integration: The integral of $x^n$ with respect to $x$ is $\frac{1}{n+1}x^{n+1}$ plus a constant of integration, provided $n \neq -1$.

6. Constant Multiple Rule: A constant can be moved in and out of the integral.

7. Simplifying Expressions: Combining like terms and simplifying expressions are algebraic techniques used to make expressions more manageable.

The solution provided follows the same step-by-step format as the original problem-solving process, including the use of substitution, differentiation, and integration rules. Each step is clearly marked and explained, and the mathematical expressions are rendered in LaTeX format for clarity.