Find the Antiderivative 2/(x^3)

$\frac{2}{x^{3}}$

Step 1

Write $\frac{2}{x^{3}}$ as a function.

$f (x) = \frac{2}{x^{3}}$

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.

$F (x) = \int \frac{2}{x^{3}} d x$

Step 4

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

$2 \int \frac{1}{x^{3}} d x$

Step 5

Apply basic rules of exponents.

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$2 \int {(x^{3})}^{- 1} d x$

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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

$2 \int x^{3 \cdot - 1} d x$

Multiply $3$ by $- 1$ .

$2 \int x^{- 3} d x$

Step 6

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

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

Step 7

Simplify the answer.

Simplify.

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

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

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Simplify.

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

Simplify.

Multiply $- 1$ by $2$ .

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

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

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

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 8

The answer is the antiderivative of the function $f (x) = \frac{2}{x^{3}}$ .

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

Find the Antiderivative 2/(x^3)

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