Find the Antiderivative f(x)=22x^21

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 $x^{21}$.

$$22(\frac{x^{22}}{22}+C)$$

Step 5:

Proceed to simplify the expression.

Step 5.1:

Express $22(\frac{x^{22}}{22}+C)$ 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:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is found by integrating $f(x)$ with respect to $x$. This is represented by $\int f(x)dx$.

2. 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.

3. 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 $n\ne -1$ and $C$ is the constant of integration.

4. Simplification: After applying the integration rules, the expression is simplified by combining like terms, canceling common factors, and performing arithmetic operations as necessary.

5. 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 $f(x)=22x^{21}$.