Find the Antiderivative f(x)=1+e^x+1/x
The given problem is asking for the antiderivative of a composite function, which is the mathematical operation that reverses the process of differentiation. The function provided includes the sum of three terms: a constant (1), an exponential function (e^x), and a reciprocal function (1/x). The request is to find a function F(x) such that its derivative, F'(x), would yield the original function f(x) = 1 + e^x + 1/x. This involves applying the rules of integration to each term separately.
$f \left(\right. x \left.\right) = 1 + e^{x} + \frac{1}{x}$
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.
$$F(x) = \int (1 + e^x + \frac{1}{x}) \, dx$$
Step 3:
Decompose the integral into a sum of simpler integrals.
$$\int dx + \int e^x \, dx + \int \frac{1}{x} \, dx$$
Step 4:
Integrate the constant term using the power rule.
$$x + C + \int e^x \, dx + \int \frac{1}{x} \, dx$$
Step 5:
Evaluate the integral of $e^x$.
$$x + C + e^x + C + \int \frac{1}{x} \, dx$$
Step 6:
Integrate $\frac{1}{x}$ to obtain the natural logarithm.
$$x + C + e^x + C + \ln(|x|) + C$$
Step 7:
Combine the terms and constants.
$$x + e^x + \ln(|x|) + C$$
Step 8:
Conclude with the antiderivative of $f(x) = 1 + e^x + \frac{1}{x}$.
$$F(x) = x + e^x + \ln(|x|) + C$$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves determining a function whose derivative is the given function. Here are the relevant knowledge points and explanations:
Indefinite Integral: The antiderivative of a function $f(x)$ is denoted by $\int f(x) \, dx$ and represents a family of functions that differ by a constant, since the derivative of a constant is zero.
Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This allows us to split a complex integral into simpler parts.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$. For the constant function $f(x) = 1$, the power rule gives us the integral $x + C$.
Exponential Function Integral: The integral of $e^x$ with respect to $x$ is $e^x + C$, as the derivative of $e^x$ is itself.
Integral of $1/x$: The integral of $\frac{1}{x}$ with respect to $x$ is $\ln(|x|) + C$, where $|x|$ denotes the absolute value of $x$. This is because the derivative of $\ln(|x|)$ is $\frac{1}{x}$ for $x \neq 0$.
Constants of Integration: When integrating, a constant of integration $C$ is added to represent the family of antiderivatives. When combining terms, multiple constants can be consolidated into a single constant.
By understanding these principles, one can systematically approach the integration of various functions to find their antiderivatives.
