Calculus Examples

$\int_{- 1}^{1} \frac{x^{2} - x}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

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

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Step 1.1.1

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

$\int_{- 1}^{1} \frac{x \cdot x - x}{x} d x$

Step 1.1.2

Factor $x$ out of $- x$ .

$\int_{- 1}^{1} \frac{x \cdot x + x \cdot - 1}{x} d x$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\int_{- 1}^{1} \frac{x (x - 1)}{x} d x$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\int_{- 1}^{1} \frac{x (x - 1)}{x} d x$

Step 1.2.2

Divide $x - 1$ by $1$ .

$\int_{- 1}^{1} x - 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{1} x d x + \int_{- 1}^{1} - 1 d x$

Step 3

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

$\frac{1}{2} x^{2}]_{- 1}^{1} + \int_{- 1}^{1} - 1 d x$

Step 4

Apply the constant rule.

$\frac{1}{2} x^{2}]_{- 1}^{1} + - x]_{- 1}^{1}$

Step 5

Simplify the answer.

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Step 5.1

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

$\frac{1}{2} x^{2} - x]_{- 1}^{1}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

One to any power is one.

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

Step 5.2.2.2

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

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

Step 5.2.2.3

Multiply $- 1$ by $1$ .

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

Step 5.2.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.2.5

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

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

Step 5.2.2.6

Combine the numerators over the common denominator.

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

Step 5.2.2.7

Simplify the numerator.

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Step 5.2.2.7.1

Multiply $- 1$ by $2$ .

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

Step 5.2.2.7.2

Subtract $2$ from $1$ .

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

Step 5.2.2.8

Move the negative in front of the fraction.

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

Step 5.2.2.9

Raise $- 1$ to the power of $2$ .

$- \frac{1}{2} - (\frac{1}{2} \cdot 1 - - 1)$

Step 5.2.2.10

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

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

Step 5.2.2.11

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2.12

Write $1$ as a fraction with a common denominator.

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

Step 5.2.2.13

Combine the numerators over the common denominator.

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

Step 5.2.2.14

Add $1$ and $2$ .

$- \frac{1}{2} - \frac{3}{2}$

Step 5.2.2.15

Combine the numerators over the common denominator.

$\frac{- 1 - 3}{2}$

Step 5.2.2.16

Subtract $3$ from $- 1$ .

$\frac{- 4}{2}$

Step 5.2.2.17

Cancel the common factor of $- 4$ and $2$ .

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Step 5.2.2.17.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2}{2}$

Step 5.2.2.17.2

Cancel the common factors.

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Step 5.2.2.17.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 2}{2 (1)}$

Step 5.2.2.17.2.2

Cancel the common factor.

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

Step 5.2.2.17.2.3

Rewrite the expression.

$\frac{- 2}{1}$

Step 5.2.2.17.2.4

Divide $- 2$ by $1$ .

$- 2$

Step 6