Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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