Find the Antiderivative f(x)=(x^3)/3
The question is asking for the antiderivative (also known as the indefinite integral) of the function f(x) = (x^3)/3. This means you need to determine the function F(x) whose derivative is f(x). The process of finding the antiderivative involves reversing the rules of differentiation to obtain a function that, when differentiated, yields the original function f(x).
$f \left(\right. x \left.\right) = \frac{x^{3}}{3}$
Answer
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 $x$.
$$\frac{1}{3} \left( \frac{x^4}{4} + C \right)$$
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.
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.
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.
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 $n \neq -1$. In this case, integrating $x^3$ gives us $\frac{x^4}{4}$.
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 $\frac{1}{12}$.
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.
