Calculus Examples

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

Step 1

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

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Step 1.1

Factor $x$ out of $d x$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

Since $\frac{1}{x^{2}}$ is constant with respect to $d$ , move $\frac{1}{x^{2}}$ out of the integral.

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{2}$ .

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

Step 4.2.2

Move $2$ to the left of $x^{2}$ .

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

Step 4.2.3

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

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