Find the Antiderivative f(x)=(3x-2)^3

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.
$F(x) = \int (3x - 2)^3 \, dx$

Step 3:

Use substitution by letting $u = 3x - 2$. Then calculate $du = 3dx$, which leads to $\frac{1}{3}du = dx$. Express the integral in terms of $u$ and $du$.

Step 3.1:

Assign $u = 3x - 2$ and find $\frac{du}{dx}$.

Step 3.1.1:

Differentiate $3x - 2$.
$\frac{d}{dx}(3x - 2)$

Step 3.1.2:

Apply the Sum Rule to find the derivative of $3x - 2$ with respect to $x$.
$\frac{d}{dx}(3x) + \frac{d}{dx}(-2)$

Step 3.1.3:

Compute $\frac{d}{dx}(3x)$.

Step 3.1.3.1:

Since $3$ is a constant, the derivative of $3x$ is $3\frac{d}{dx}(x)$.
$3\frac{d}{dx}(x) + \frac{d}{dx}(-2)$

Step 3.1.3.2:

Use the Power Rule, which states the derivative of $x^n$ is $nx^{n-1}$ where $n = 1$.
$3 \cdot 1 + \frac{d}{dx}(-2)$

Step 3.1.3.3:

Multiply $3$ by $1$.
$3 + \frac{d}{dx}(-2)$

Step 3.1.4:

Apply the Constant Rule.

Step 3.1.4.1:

Since $-2$ is a constant, its derivative is $0$.
$3 + 0$

Step 3.1.4.2:

Combine $3$ and $0$.
$3$

Step 3.2:

Reformulate the integral using $u$ and $du$.
$\int u^3 \frac{1}{3} du$

Step 4:

Combine $u^3$ and $\frac{1}{3}$.
$\int \frac{u^3}{3} du$

Step 5:

Since $\frac{1}{3}$ is a constant, move it outside the integral.
$\frac{1}{3} \int u^3 du$

Step 6:

Integrate $u^3$ using the Power Rule to get $\frac{1}{4}u^4$.
$\frac{1}{3} \left( \frac{1}{4}u^4 + C \right)$

Step 7:

Simplify the expression.

Step 7.1:

Express $\frac{1}{3} \left( \frac{1}{4}u^4 + C \right)$ as $\frac{1}{3} \cdot \frac{1}{4} u^4 + C$.
$\frac{1}{3} \cdot \frac{1}{4} u^4 + C$

Step 7.2:

Simplify further.

Step 7.2.1:

Multiply $\frac{1}{3}$ by $\frac{1}{4}$.
$\frac{1}{3 \cdot 4} u^4 + C$

Step 7.2.2:

Calculate $3 \cdot 4$.
$\frac{1}{12} u^4 + C$

Step 8:

Substitute back the original variable by replacing $u$ with $3x - 2$.
$\frac{1}{12} ((3x - 2)^4) + C$

Step 9:

Conclude with the antiderivative of $f(x) = (3x - 2)^3$.
$F(x) = \frac{1}{12} ((3x - 2)^4) + C$

Knowledge Notes:

The problem involves finding the antiderivative (indefinite integral) of a given function. The process requires the following knowledge:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is represented as $F(x) = \int f(x) \, dx$, where $F'(x) = f(x)$.

2. Substitution Rule: This is a method for evaluating integrals. If $u=g(x)$ is a differentiable function whose range is an interval $I$ and $f$ is continuous on $I$, then $\int f(g(x))g'(x) \, dx = \int f(u) \, du$.

3. Sum Rule for Derivatives: The derivative of a sum of functions is the sum of their derivatives, i.e., $(f + g)' = f' + g'$.

4. Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function, i.e., $(cf)' = cf'$.

5. Power Rule for Derivatives: If $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

6. Constant Rule for Derivatives: The derivative of a constant is zero.

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

8. Constants in Integration: Constants can be factored out of an integral.

Using these principles, the problem-solving process involves substituting $u = 3x - 2$ to simplify the integral, finding the derivative $du$, and then integrating with respect to $u$ before substituting back to express the antiderivative in terms of $x$.