Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

$x]_{- 1}^{1} - \int_{- 1}^{1} | x | d x$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Substitute and simplify.

Tap for more steps...

Step 10.2.1

Evaluate $x$ at $1$ and at $- 1$ .

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

Step 10.2.2

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

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

Step 10.2.3

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

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

Step 10.2.4

Simplify.

Tap for more steps...

Step 10.2.4.1

Add $1$ and $1$ .

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

Step 10.2.4.2

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

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

Step 10.2.4.3

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

Tap for more steps...

Step 10.2.4.3.1

Factor $2$ out of $0$ .

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

Step 10.2.4.3.2

Cancel the common factors.

Tap for more steps...

Step 10.2.4.3.2.1

Factor $2$ out of $2$ .

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

Step 10.2.4.3.2.2

Cancel the common factor.

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

Step 10.2.4.3.2.3

Rewrite the expression.

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

Step 10.2.4.3.2.4

Divide $0$ by $1$ .

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

Step 10.2.4.4

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

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

Step 10.2.4.5

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

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

Step 10.2.4.6

Multiply $- 1$ by $- 1$ .

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

Step 10.2.4.7

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

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

Step 10.2.4.8

One to any power is one.

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

Step 10.2.4.9

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

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

Step 10.2.4.10

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

Tap for more steps...

Step 10.2.4.10.1

Factor $2$ out of $0$ .

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

Step 10.2.4.10.2

Cancel the common factors.

Tap for more steps...

Step 10.2.4.10.2.1

Factor $2$ out of $2$ .

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

Step 10.2.4.10.2.2

Cancel the common factor.

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

Step 10.2.4.10.2.3

Rewrite the expression.

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

Step 10.2.4.10.2.4

Divide $0$ by $1$ .

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

Step 10.2.4.11

Multiply $- 1$ by $0$ .

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

Step 10.2.4.12

Add $\frac{1}{2}$ and $0$ .

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

Step 10.2.4.13

Combine the numerators over the common denominator.

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

Step 10.2.4.14

Add $1$ and $1$ .

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

Step 10.2.4.15

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.4.15.1

Cancel the common factor.

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

Step 10.2.4.15.2

Rewrite the expression.

$2 - 1 \cdot 1$

Step 10.2.4.16

Multiply $- 1$ by $1$ .

$2 - 1$

Step 10.2.4.17

Subtract $1$ from $2$ .

$1$

Step 11