Find the Antiderivative f(x)=22x^21
The given problem is asking to perform a mathematical operation called integration on a specific function, f(x)=22x^21. Integration is the reverse process of differentiation and is used to find antiderivatives or indefinite integrals. The goal here is to find a new function F(x) whose derivative will equal the given function f(x). The process involves applying integral calculus rules according to the power of x to find this antiderivative.
$f \left(\right. x \left.\right) = 22 x^{21}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step 2:
Prepare to integrate the function.
$$F(x) = \int 22x^{21} \, dx$$
Step 3:
Extract the constant coefficient $22$ from the integral.
$$22 \int x^{21} \, dx$$
Step 4:
Apply the Power Rule for integration to $x^{21}$.
$$22 \left( \frac{x^{22}}{22} + C \right)$$
Step 5:
Proceed to simplify the expression.
Step 5.1:
Express $22 \left( \frac{x^{22}}{22} + C \right)$ as $22 \cdot \frac{x^{22}}{22} + C$.
$$22 \cdot \frac{x^{22}}{22} + C$$
Step 5.2:
Perform the simplification.
Step 5.2.1:
Combine the $22$ and $\frac{1}{22}$.
$$\frac{22}{22} x^{22} + C$$
Step 5.2.2:
Eliminate the common factors.
Step 5.2.2.1:
Remove the common factor.
$$\frac{\cancel{22}}{\cancel{22}} x^{22} + C$$
Step 5.2.2.2:
Rewrite the simplified expression.
$$x^{22} + C$$
Step 5.2.3:
Multiply $x^{22}$ by $1$.
$$x^{22} + C$$
Step 6:
Conclude with the antiderivative of $f(x) = 22x^{21}$.
$$F(x) = x^{22} + C$$
Knowledge Notes:
To solve for the antiderivative of a function, we follow a standard process that involves integrating the given function. Here are the relevant knowledge points and detailed explanations:
Indefinite Integral: The antiderivative of a function $f(x)$ is found by integrating $f(x)$ with respect to $x$. This is represented by $\int f(x) \, dx$.
Constant Multiple Rule: When a constant is multiplied by a function, it can be factored out of the integral. For example, $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.
Power Rule for Integration: The Power Rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.
Simplification: After applying the integration rules, the expression is simplified by combining like terms, canceling common factors, and performing arithmetic operations as necessary.
Constant of Integration: Since the integral is indefinite, a constant of integration $C$ is added to the result to account for all possible antiderivatives.
Using these principles, we can systematically find the antiderivative of a given polynomial function like $f(x) = 22x^{21}$.
