Find the Antiderivative 2/(x^2)

$\frac{2}{x^{2}}$

Step 1

Write $\frac{2}{x^{2}}$ as a function.

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

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.

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

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5

Apply basic rules of exponents.

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Step 6

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

$2 (- x^{- 1} + C)$

Step 7

Simplify the answer.

Rewrite $2 (- x^{- 1} + C)$ as $2 (- \frac{1}{x}) + C$ .

$2 (- \frac{1}{x}) + C$

Simplify.

Multiply $- 1$ by $2$ .

$- 2 \frac{1}{x} + C$

Combine $- 2$ and $\frac{1}{x}$ .

$\frac{- 2}{x} + C$

Move the negative in front of the fraction.

$- \frac{2}{x} + C$

Step 8

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

$F (x) =$ $- \frac{2}{x} + C$