Calculus Examples

$f (x) = \cos (x) - 1$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \cos (x) - 1 d x$

Step 3

Split the single integral into multiple integrals.

$\int \cos (x) d x + \int - 1 d x$

Step 4

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$\sin (x) + C + \int - 1 d x$

Step 5

Apply the constant rule.

$\sin (x) + C - x + C$

Step 6

Simplify.

$\sin (x) - x + C$

Step 7

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

$F (x) =$ $\sin (x) - x + C$