Calculus Examples

$(1 - \csc (x) \cot (x))$

Step 1

Write $(1 - \csc (x) \cot (x))$ as a function.

$f (x) = (1 - \csc (x) \cot (x))$

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.

$F (x) = \int 1 - \csc (x) \cot (x) d x$

Step 4

Remove parentheses.

$\int 1 - \csc (x) \cot (x) d x$

Step 5

Split the single integral into multiple integrals.

$\int d x + \int - \csc (x) \cot (x) d x$

Step 6

Apply the constant rule.

$x + C + \int - \csc (x) \cot (x) d x$

Step 7

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

$x + C - \int \csc (x) \cot (x) d x$

Step 8

Since the derivative of $- \csc (x)$ is $\csc (x) \cot (x)$ , the integral of $\csc (x) \cot (x)$ is $- \csc (x)$ .

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

Step 9

Simplify.

$x + \csc (x) + C$

Step 10

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

$F (x) =$ $x + \csc (x) + C$