Calculus Examples

$\int_{- 2}^{2} | x^{2} - 4 | d x$

Step 1

Split up the integral depending on where $x^{2} - 4$ is positive and negative.

$\int_{- 2}^{2} - x^{2} + 4 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{2} - x^{2} d x + \int_{- 2}^{2} 4 d x$

Step 3

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

$- \int_{- 2}^{2} x^{2} d x + \int_{- 2}^{2} 4 d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{1}{3} x^{3}]_{- 2}^{2}) + \int_{- 2}^{2} 4 d x$

Step 5

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{- 2}^{2}) + \int_{- 2}^{2} 4 d x$

Step 6

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- 2}^{2}) + 4 x]_{- 2}^{2}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 2$ .

$- ((\frac{2^{3}}{3}) - \frac{{(- 2)}^{3}}{3}) + 4 x]_{- 2}^{2}$

Step 7.2

Evaluate $4 x$ at $2$ and at $- 2$ .

$- (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3

Simplify.

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Step 7.3.1

Raise $2$ to the power of $3$ .

$- (\frac{8}{3} - \frac{{(- 2)}^{3}}{3}) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.2

Raise $- 2$ to the power of $3$ .

$- (\frac{8}{3} - \frac{- 8}{3}) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.3

Move the negative in front of the fraction.

$- (\frac{8}{3} - - \frac{8}{3}) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.4

Multiply $- 1$ by $- 1$ .

$- (\frac{8}{3} + 1 (\frac{8}{3})) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.5

Multiply $\frac{8}{3}$ by $1$ .

$- (\frac{8}{3} + \frac{8}{3}) + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.6

Combine the numerators over the common denominator.

$- \frac{8 + 8}{3} + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.7

Add $8$ and $8$ .

$- \frac{16}{3} + 4 \cdot 2 - 4 \cdot - 2$

Step 7.3.8

Multiply $4$ by $2$ .

$- \frac{16}{3} + 8 - 4 \cdot - 2$

Step 7.3.9

Multiply $- 4$ by $- 2$ .

$- \frac{16}{3} + 8 + 8$

Step 7.3.10

Add $8$ and $8$ .

$- \frac{16}{3} + 16$

Step 7.3.11

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{16}{3} + 16 \cdot \frac{3}{3}$

Step 7.3.12

Combine $16$ and $\frac{3}{3}$ .

$- \frac{16}{3} + \frac{16 \cdot 3}{3}$

Step 7.3.13

Combine the numerators over the common denominator.

$\frac{- 16 + 16 \cdot 3}{3}$

Step 7.3.14

Simplify the numerator.

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Step 7.3.14.1

Multiply $16$ by $3$ .

$\frac{- 16 + 48}{3}$

Step 7.3.14.2

Add $- 16$ and $48$ .

$\frac{32}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3}$

Decimal Form:

$10.\overline{6}$

Mixed Number Form:

$10 \frac{2}{3}$

Step 9