Calculus Examples

\frac{x^{3} - x}{x - 1}

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor

x

$x$ out of

x^{3} - x

$x^{3} - x$ .

Tap for more steps...

Step 1.1.1

Factor

x

$x$ out of

x^{3}

$x^{3}$ .

\frac{x \cdot x^{2} - x}{x - 1}

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

Step 1.1.2

Factor

x

$x$ out of

- x

$- x$ .

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

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

Step 1.1.3

Factor

x

$x$ out of

x \cdot x^{2} + x \cdot - 1

$x \cdot x^{2} + x \cdot - 1$ .

\frac{x (x^{2} - 1)}{x - 1}

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

\frac{x (x^{2} - 1)}{x - 1}

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

Step 1.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\frac{x (x^{2} - 1^{2})}{x - 1}

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

\frac{x (x + 1) (x - 1)}{x - 1}

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

\frac{x (x + 1) (x - 1)}{x - 1}

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Cancel the common factor of

x - 1

$x - 1$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

\frac{x (x + 1) (x - 1)}{x - 1}

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

Step 2.1.2

Divide

x (x + 1)

$x (x + 1)$ by

1

$1$ .

x (x + 1)

$x (x + 1)$

x (x + 1)

$x (x + 1)$

Step 2.2

Apply the distributive property.

x \cdot x + x \cdot 1

$x \cdot x + x \cdot 1$

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot 1

$x^{2} + x \cdot 1$

Step 2.3.2

Multiply

x

$x$ by

1

$1$ .

x^{2} + x

$x^{2} + x$

x^{2} + x

$x^{2} + x$

x^{2} + x

$x^{2} + x$

\frac{x^{3} - x}{x - 1}

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

(

(

)

)

|

|

[

[

]

]

√

√





≥

≥





∫

∫













7

7

8

8

9

9













≤

≤





°

°

θ

θ









4

4

5

5

6

6

/

/

^

^

×

×

>

>





π

π













1

1

2

2

3

3

-

-

+

+

÷

÷

<

<





∞

∞





!

!



,

,

0

0

.

.

%

%



=

=









[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$