Calculus Examples

$\int_{- 1}^{1} [(2 - x^{2}) - (x^{2})] d x$

Step 1

Remove parentheses.

$\int_{- 1}^{1} (2 - x^{2}) - (x^{2}) d x$

Step 2

Subtract $x^{2}$ from $- x^{2}$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $2 x$ at $1$ and at $- 1$ .

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

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 1$ .

$2 \cdot 1 - 2 \cdot - 1 - 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $2$ by $1$ .

$2 - 2 \cdot - 1 - 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 7.2.3.2

Multiply $- 2$ by $- 1$ .

$2 + 2 - 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 7.2.3.3

Add $2$ and $2$ .

$4 - 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 7.2.3.4

One to any power is one.

$4 - 2 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3})$

Step 7.2.3.5

Raise $- 1$ to the power of $3$ .

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

Step 7.2.3.6

Move the negative in front of the fraction.

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

Step 7.2.3.7

Multiply $- 1$ by $- 1$ .

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

Step 7.2.3.8

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

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

Step 7.2.3.9

Combine the numerators over the common denominator.

$4 - 2 \frac{1 + 1}{3}$

Step 7.2.3.10

Add $1$ and $1$ .

$4 - 2 (\frac{2}{3})$

Step 7.2.3.11

Combine $- 2$ and $\frac{2}{3}$ .

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

Step 7.2.3.12

Multiply $- 2$ by $2$ .

$4 + \frac{- 4}{3}$

Step 7.2.3.13

Move the negative in front of the fraction.

$4 - \frac{4}{3}$

Step 7.2.3.14

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \cdot \frac{3}{3} - \frac{4}{3}$

Step 7.2.3.15

Combine $4$ and $\frac{3}{3}$ .

$\frac{4 \cdot 3}{3} - \frac{4}{3}$

Step 7.2.3.16

Combine the numerators over the common denominator.

$\frac{4 \cdot 3 - 4}{3}$

Step 7.2.3.17

Simplify the numerator.

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Step 7.2.3.17.1

Multiply $4$ by $3$ .

$\frac{12 - 4}{3}$

Step 7.2.3.17.2

Subtract $4$ from $12$ .

$\frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$

Step 9