Find the Antiderivative f(x)=x^11

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.

$$F(x) = \int x^{11} \, dx$$

Step 3:

Apply the Power Rule for Integration, which states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$, plus a constant of integration $C$.

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

Step 4:

Conclude with the antiderivative of the function $f(x) = x^{11}$.

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

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the process of differentiation. The Power Rule for Integration is a fundamental technique used when integrating powers of $x$. It states that for any real number $n \neq -1$, the indefinite integral of $x^n$ with respect to $x$ is given by:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

where $C$ is the constant of integration, representing an arbitrary constant that can take any value. This constant is necessary because differentiation of a constant yields zero, and thus the original function could have had any constant value added to it.

When applying the Power Rule, it is important to increment the exponent by one and then divide by the new exponent. The constant of integration is always added to the result of an indefinite integral to account for all possible antiderivatives.

In the given problem, the function to integrate is $f(x) = x^{11}$. Applying the Power Rule, we increase the exponent by one to get $12$ and then divide by this new exponent. This results in the antiderivative $\frac{x^{12}}{12} + C$.