Calculus Examples

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

Step 1

Split up the integral depending on where $x$ is positive and negative.

$\int_{- 1}^{0} - x d x + \int_{0}^{2} x d x$

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{- 1}^{0} x d x + \int_{0}^{2} x d x$

Step 3

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

$- (\frac{1}{2} x^{2}]_{- 1}^{0}) + \int_{0}^{2} x d x$

Step 4

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

$- (\frac{x^{2}}{2}]_{- 1}^{0}) + \int_{0}^{2} x d x$

Step 5

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

$- (\frac{x^{2}}{2}]_{- 1}^{0}) + \frac{1}{2} x^{2}]_{0}^{2}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 1$ .

$- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + \frac{1}{2} x^{2}]_{0}^{2}$

Step 6.2

Evaluate $\frac{1}{2} x^{2}$ at $2$ and at $0$ .

$- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3

Simplify.

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Step 6.3.1

Raising $0$ to any positive power yields $0$ .

$- (\frac{0}{2} - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.2

Cancel the common factor of $0$ and $2$ .

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Step 6.3.2.1

Factor $2$ out of $0$ .

$- (\frac{2 (0)}{2} - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $2$ out of $2$ .

$- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.2.2.2

Cancel the common factor.

$- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.2.2.3

Rewrite the expression.

$- (\frac{0}{1} - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.2.2.4

Divide $0$ by $1$ .

$- (0 - \frac{{(- 1)}^{2}}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.3

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

$- (0 - \frac{1}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.4

Subtract $\frac{1}{2}$ from $0$ .

$- - \frac{1}{2} + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.5

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.6

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

$\frac{1}{2} + (\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.3.7

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

$\frac{1}{2} + \frac{1}{2} \cdot 4 - \frac{1}{2} \cdot 0^{2}$

Step 6.3.8

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2} + \frac{4}{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.3.9

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

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Step 6.3.9.1

Factor $2$ out of $4$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.3.9.2

Cancel the common factors.

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Step 6.3.9.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2 (1)} - \frac{1}{2} \cdot 0^{2}$

Step 6.3.9.2.2

Cancel the common factor.

$\frac{1}{2} + \frac{2 \cdot 2}{2 \cdot 1} - \frac{1}{2} \cdot 0^{2}$

Step 6.3.9.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{2}{1} - \frac{1}{2} \cdot 0^{2}$

Step 6.3.9.2.4

Divide $2$ by $1$ .

$\frac{1}{2} + 2 - \frac{1}{2} \cdot 0^{2}$

Step 6.3.10

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} + 2 - \frac{1}{2} \cdot 0$

Step 6.3.11

Multiply $0$ by $- 1$ .

$\frac{1}{2} + 2 + 0 (\frac{1}{2})$

Step 6.3.12

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{2} + 2 + 0$

Step 6.3.13

Add $2$ and $0$ .

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

Step 6.3.14

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

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

Step 6.3.15

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

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

Step 6.3.16

Combine the numerators over the common denominator.

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

Step 6.3.17

Simplify the numerator.

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Step 6.3.17.1

Multiply $2$ by $2$ .

$\frac{1 + 4}{2}$

Step 6.3.17.2

Add $1$ and $4$ .

$\frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$

Step 8