Find the Antiderivative f(x)=23e^x
The problem asks you to calculate the antiderivative (also known as the indefinite integral) of the function f(x) = 23e^x. In simple terms, it's looking for the function F(x) whose derivative is 23e^x. An antiderivative is a reverse process of differentiation and is part of integral calculus. The problem requires knowledge of integration rules and properties of exponential functions.
$f \left(\right. x \left.\right) = 23 e^{x}$
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.
$$ F(x) = \int 23e^x \, dx $$
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:
Antiderivative: An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$. The process of finding $F(x)$ is called integration.
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.
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.
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.
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 $F(x) = 23e^x + C$.
