Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

$\int \frac{\cos (x)}{x} + \frac{- 1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{\cos (x)}{x} d x + \int \frac{- 1}{x} d x$

Step 3

Move the negative in front of the fraction.

$\int \frac{\cos (x)}{x} d x + \int - \frac{1}{x} d x$

Step 4

$\int \frac{\cos (x)}{x} d x$ is a special integral. The integral is the cosine integral function.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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