Find the Antiderivative f(x)=23e^x

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.

$$F(x)=\int 23e^{x}dx$$

Step 3:

Extract the constant factor $23$ from the integral as it does not depend on $x$.

$$23\int e^{x}dx$$

Step 4:

Compute the integral of $e^x$, which is itself.

$$23(e^x+C)$$

Step 5:

Combine the constant multiple with the integral result.

$$23e^x+C$$

Step 6:

Conclude with the antiderivative of $f(x)=23e^x$.

$$F(x)=23e^x+C$$

Knowledge Notes:

The process of finding an antiderivative involves integrating a given function. Here are the relevant knowledge points for this problem:

1. Antiderivative: An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F^{\prime }(x)=f(x)$. The process of finding $F(x)$ is called integration.

2. Indefinite Integral: The indefinite integral, denoted by $\int f(x)dx$, represents the collection of all antiderivatives of $f(x)$. It includes a constant of integration $C$, since the derivative of a constant is zero.

3. Constants in Integration: When integrating a function multiplied by a constant, the constant can be factored out of the integral. This is due to the linearity of integration.

4. Integral of Exponential Functions: The integral of $e^x$ with respect to $x$ is $e^x$, as the rate of change of the exponential function is proportional to the function itself.

5. Constant of Integration: When finding an indefinite integral, there is an arbitrary constant $C$ added to the result. This constant accounts for the fact that there are infinitely many antiderivatives, each differing by a constant.

By applying these principles, the antiderivative of $f(x)=23e^x$ is found to be $F(x)=23e^x+C$.