Find the Antiderivative f(x)=1+e^x+1/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.

$$F(x)=\int (1+e^x+\frac{1}{x})dx$$

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:

1. 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.

2. 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.

3. Power Rule for Integration: For any real number $n\ne -1$, 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 $x+C$.

4. Exponential Function Integral: The integral of $e^x$ with respect to $x$ is $e^x+C$, as the derivative of $e^x$ is itself.

5. 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 $x\ne 0$.

6. 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.