Calculus Examples

$\int_{- 2}^{2} 12 - 3 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{2} 12 d x + \int_{- 2}^{2} - 3 x^{2} d x$

Step 2

Apply the constant rule.

$12 x]_{- 2}^{2} + \int_{- 2}^{2} - 3 x^{2} d x$

Step 3

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$12 x]_{- 2}^{2} - 3 \int_{- 2}^{2} x^{2} d x$

Step 4

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

$12 x]_{- 2}^{2} - 3 (\frac{1}{3} x^{3}]_{- 2}^{2})$

Step 5

Simplify the answer.

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Step 5.1

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

$12 x]_{- 2}^{2} - 3 (\frac{x^{3}}{3}]_{- 2}^{2})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $12 x$ at $2$ and at $- 2$ .

$(12 \cdot 2) - 12 \cdot - 2 - 3 (\frac{x^{3}}{3}]_{- 2}^{2})$

Step 5.2.2

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

$12 \cdot 2 - 12 \cdot - 2 - 3 (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $12$ by $2$ .

$24 - 12 \cdot - 2 - 3 (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.2

Multiply $- 12$ by $- 2$ .

$24 + 24 - 3 (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.3

Add $24$ and $24$ .

$48 - 3 (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.4

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

$48 - 3 (\frac{8}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.5

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

$48 - 3 (\frac{8}{3} - \frac{- 8}{3})$

Step 5.2.3.6

Move the negative in front of the fraction.

$48 - 3 (\frac{8}{3} - - \frac{8}{3})$

Step 5.2.3.7

Multiply $- 1$ by $- 1$ .

$48 - 3 (\frac{8}{3} + 1 (\frac{8}{3}))$

Step 5.2.3.8

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

$48 - 3 (\frac{8}{3} + \frac{8}{3})$

Step 5.2.3.9

Combine the numerators over the common denominator.

$48 - 3 \frac{8 + 8}{3}$

Step 5.2.3.10

Add $8$ and $8$ .

$48 - 3 (\frac{16}{3})$

Step 5.2.3.11

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

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

Step 5.2.3.12

Multiply $- 3$ by $16$ .

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

Step 5.2.3.13

Cancel the common factor of $- 48$ and $3$ .

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Step 5.2.3.13.1

Factor $3$ out of $- 48$ .

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

Step 5.2.3.13.2

Cancel the common factors.

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Step 5.2.3.13.2.1

Factor $3$ out of $3$ .

$48 + \frac{3 \cdot - 16}{3 (1)}$

Step 5.2.3.13.2.2

Cancel the common factor.

$48 + \frac{3 \cdot - 16}{3 \cdot 1}$

Step 5.2.3.13.2.3

Rewrite the expression.

$48 + \frac{- 16}{1}$

Step 5.2.3.13.2.4

Divide $- 16$ by $1$ .

$48 - 16$

Step 5.2.3.14

Subtract $16$ from $48$ .

$32$

Step 6