Calculus Examples

$e^{x} + \frac{1}{x}$

Step 1

Write $e^{x} + \frac{1}{x}$ as a function.

$f (x) = e^{x} + \frac{1}{x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

Split the single integral into multiple integrals.

$\int e^{x} d x + \int \frac{1}{x} d x$

Step 5

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$e^{x} + C + \int \frac{1}{x} d x$

Step 6

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{x} + C + \ln (| x |) + C$

Step 7

Simplify.

$e^{x} + \ln (| x |) + C$

Step 8

The answer is the antiderivative of the function $f (x) = e^{x} + \frac{1}{x}$ .

$F (x) =$ $e^{x} + \ln (| x |) + C$