Calculus Examples

$f (x) = \frac{31}{\sin^{2} (x)}$

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.

$F (x) = \int \frac{31}{\sin^{2} (x)} d x$

Step 3

Since $31$ is constant with respect to $x$ , move $31$ out of the integral.

$31 \int \frac{1}{\sin^{2} (x)} d x$

Step 4

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

$31 \int \csc^{2} (x) d x$

Step 5

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$31 (- \cot (x) + C)$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

$31 (- \cot (x)) + C$

Step 6.2

Multiply $- 1$ by $31$ .

$- 31 \cot (x) + C$

Step 7

The answer is the antiderivative of the function $f (x) = \frac{31}{\sin^{2} (x)}$ .

$F (x) =$ $- 31 \cot (x) + C$