Calculus Examples

$\int_{- 1}^{1} 3 | x^{3} | d x$

Step 1

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

$3 \int_{- 1}^{1} | x^{3} | d x$

Step 2

Split up the integral depending on where $x^{3}$ is positive and negative.

$3 (\int_{- 1}^{0} - x^{3} d x + \int_{0}^{1} x^{3} d x)$

Step 3

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

$3 (- \int_{- 1}^{0} x^{3} d x + \int_{0}^{1} x^{3} d x)$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$3 (- (\frac{1}{4} x^{4}]_{- 1}^{0}) + \int_{0}^{1} x^{3} d x)$

Step 5

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (- (\frac{x^{4}}{4}]_{- 1}^{0}) + \int_{0}^{1} x^{3} d x)$

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$3 (- (\frac{x^{4}}{4}]_{- 1}^{0}) + \frac{1}{4} x^{4}]_{0}^{1})$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (- (\frac{x^{4}}{4}]_{- 1}^{0}) + \frac{x^{4}}{4}]_{0}^{1})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{4}}{4}$ at $0$ and at $- 1$ .

$3 (- ((\frac{0^{4}}{4}) - \frac{{(- 1)}^{4}}{4}) + \frac{x^{4}}{4}]_{0}^{1})$

Step 7.2.2

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$3 (- ((\frac{0^{4}}{4}) - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

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

$3 (- (\frac{0}{4} - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.2

Cancel the common factor of $0$ and $4$ .

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Step 7.2.3.2.1

Factor $4$ out of $0$ .

$3 (- (\frac{4 (0)}{4} - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.2.2

Cancel the common factors.

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Step 7.2.3.2.2.1

Factor $4$ out of $4$ .

$3 (- (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.2.2.2

Cancel the common factor.

$3 (- (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.2.2.3

Rewrite the expression.

$3 (- (\frac{0}{1} - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.2.2.4

Divide $0$ by $1$ .

$3 (- (0 - \frac{{(- 1)}^{4}}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.3

Raise $- 1$ to the power of $4$ .

$3 (- (0 - \frac{1}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.4

Subtract $\frac{1}{4}$ from $0$ .

$3 (- - \frac{1}{4} + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.5

Multiply $- 1$ by $- 1$ .

$3 (1 (\frac{1}{4}) + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.6

Multiply $\frac{1}{4}$ by $1$ .

$3 (\frac{1}{4} + (\frac{1^{4}}{4}) - \frac{0^{4}}{4})$

Step 7.2.3.7

One to any power is one.

$3 (\frac{1}{4} + \frac{1}{4} - \frac{0^{4}}{4})$

Step 7.2.3.8

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

$3 (\frac{1}{4} + \frac{1}{4} - \frac{0}{4})$

Step 7.2.3.9

Cancel the common factor of $0$ and $4$ .

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Step 7.2.3.9.1

Factor $4$ out of $0$ .

$3 (\frac{1}{4} + \frac{1}{4} - \frac{4 (0)}{4})$

Step 7.2.3.9.2

Cancel the common factors.

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Step 7.2.3.9.2.1

Factor $4$ out of $4$ .

$3 (\frac{1}{4} + \frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 7.2.3.9.2.2

Cancel the common factor.

$3 (\frac{1}{4} + \frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 7.2.3.9.2.3

Rewrite the expression.

$3 (\frac{1}{4} + \frac{1}{4} - \frac{0}{1})$

Step 7.2.3.9.2.4

Divide $0$ by $1$ .

$3 (\frac{1}{4} + \frac{1}{4} - 0)$

Step 7.2.3.10

Multiply $- 1$ by $0$ .

$3 (\frac{1}{4} + \frac{1}{4} + 0)$

Step 7.2.3.11

Add $\frac{1}{4}$ and $0$ .

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

Step 7.2.3.12

Combine the numerators over the common denominator.

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

Step 7.2.3.13

Add $1$ and $1$ .

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

Step 7.2.3.14

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

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Step 7.2.3.14.1

Factor $2$ out of $2$ .

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

Step 7.2.3.14.2

Cancel the common factors.

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Step 7.2.3.14.2.1

Factor $2$ out of $4$ .

$3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.3.14.2.2

Cancel the common factor.

$3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.3.14.2.3

Rewrite the expression.

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

Step 7.2.3.15

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$

Step 9