Calculus Examples

$f (x) = - \frac{1}{x^{2}}$

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.

$F (x) = \int - \frac{1}{x^{2}} d x$

Step 3

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

$- \int \frac{1}{x^{2}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$- \int {(x^{2})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \int x^{2 \cdot - 1} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int x^{- 2} d x$

Step 5

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$- (- x^{- 1} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $- (- x^{- 1} + C)$ as $- - \frac{1}{x} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{x} + C$

Step 6.2.2

Multiply $\frac{1}{x}$ by $1$ .

$\frac{1}{x} + C$

Step 7

The answer is the antiderivative of the function $f (x) = - \frac{1}{x^{2}}$ .

$F (x) =$ $\frac{1}{x} + C$