Calculus Examples

$\frac{x - 1}{x}$

Step 1

Write $\frac{x - 1}{x}$ as a function.

$f (x) = \frac{x - 1}{x}$

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.

$F (x) = \int \frac{x - 1}{x} d x$

Step 4

Split the fraction into multiple fractions.

$\int \frac{x}{x} + \frac{- 1}{x} d x$

Step 5

Split the single integral into multiple integrals.

$\int \frac{x}{x} d x + \int \frac{- 1}{x} d x$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $x$ .

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Step 6.1.1

Cancel the common factor.

$\int \frac{x}{x} d x + \int \frac{- 1}{x} d x$

Step 6.1.2

Rewrite the expression.

$\int d x + \int \frac{- 1}{x} d x$

Step 6.2

Move the negative in front of the fraction.

$\int d x + \int - \frac{1}{x} d x$

Step 7

Apply the constant rule.

$x + C + \int - \frac{1}{x} d x$

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$x + C - \int \frac{1}{x} d x$

Step 9

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

$x + C - (\ln (| x |) + C)$

Step 10

Simplify.

$x - \ln (| x |) + C$

Step 11

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

$F (x) =$ $x - \ln (| x |) + C$