Find the Antiderivative f(x)=(x^3)/3

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$$F(x)=\int f(x)dx$$

Step 2:

Write down the integral that needs to be solved.

$$F(x)=\int \frac{x^3}{3}dx$$

Step 3:

Extract the constant $\frac{1}{3}$ from the integral as it does not depend on $x$.

$$\frac{1}{3}\int x^{3}dx$$

Step 4:

Apply the Power Rule for integration to find the integral of $x^3$ with respect to $x$.

$$\frac{1}{3}(\frac{x^4}{4}+C)$$

Step 5:

Begin simplifying the expression.

Step 5.1:

Rewrite the expression by distributing the constant $\frac{1}{3}$.

$$\frac{1}{3}\cdot \frac{x^4}{4}+C$$

Step 5.2:

Continue simplification.

Step 5.2.1:

Combine the fractions $\frac{1}{3}$ and $\frac{1}{4}$.

$$\frac{1}{3\cdot 4}{x}^4+C$$

Step 5.2.2:

Calculate the product of $3$ and $4$.

$$\frac{1}{12}{x}^4+C$$

Step 6:

Conclude with the antiderivative of the function $f(x)=\frac{x^3}{3}$.

$$F(x)=\frac{1}{12}{x}^4+C$$

Knowledge Notes:

The process of finding the antiderivative or the indefinite integral of a function involves reversing the process of differentiation. In this case, we are given a function $f(x)=\frac{x^3}{3}$ and asked to find its antiderivative $F(x)$. The steps taken are based on the fundamental principles of calculus and the rules of integration.

1. Indefinite Integral: The antiderivative of a function is represented by the indefinite integral. It is written as $\int f(x)dx$ and includes an arbitrary constant $C$ since the derivative of a constant is zero.

2. Constant Multiple Rule: When a constant is multiplied by a function, the integral of the product is the constant multiplied by the integral of the function. This is expressed as $\int k\cdot f(x)dx=k\cdot \int f(x)dx$, where $k$ is a constant.

3. Power Rule for Integration: This rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided that $n\ne -1$. In this case, integrating $x^3$ gives us $\frac{x^4}{4}$.

4. Combining Constants: When simplifying expressions involving constants, we can combine them by performing arithmetic operations. For example, $\frac{1}{3}\cdot \frac{1}{4}$ is simplified to $\frac{1}{12}$.

5. Arbitrary Constant of Integration: When finding the indefinite integral, we add a constant $C$ because the derivative of a constant is zero, and thus, it does not affect the differentiation process. This constant accounts for all possible vertical shifts of the antiderivative function.