Calculus Examples

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

Step 1

Write $\frac{1}{x^{3}} d x$ as a function.

$f (x) = \frac{1}{x^{3}} d x$

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.

$F (x) = \int \frac{1}{x^{3}} \cdot (d x) d x$

Step 4

Simplify.

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Step 4.1

Combine $d$ and $\frac{1}{x^{3}}$ .

$\int \frac{d}{x^{3}} \cdot x d x$

Step 4.2

Combine $\frac{d}{x^{3}}$ and $x$ .

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

Step 4.3

Cancel the common factor of $x$ and $x^{3}$ .

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Step 4.3.1

Factor $x$ out of $d x$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $x$ out of $x^{3}$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Combine $d$ and $\frac{1}{x}$ .

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

Step 9

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{3}} \cdot (d x)$ .

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