Calculus Examples

$\int_{- 3}^{0} - 12 x - x^{2} + x^{3} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 3}^{0} - 12 x d x + \int_{- 3}^{0} - x^{2} d x + \int_{- 3}^{0} x^{3} d x$

Step 2

Since $- 12$ is constant with respect to $x$ , move $- 12$ out of the integral.

$- 12 \int_{- 3}^{0} x d x + \int_{- 3}^{0} - x^{2} d x + \int_{- 3}^{0} x^{3} d x$

Step 3

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

$- 12 (\frac{1}{2} x^{2}]_{- 3}^{0}) + \int_{- 3}^{0} - x^{2} d x + \int_{- 3}^{0} x^{3} d x$

Step 4

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

$- 12 (\frac{x^{2}}{2}]_{- 3}^{0}) + \int_{- 3}^{0} - x^{2} d x + \int_{- 3}^{0} x^{3} d x$

Step 5

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

$- 12 (\frac{x^{2}}{2}]_{- 3}^{0}) - \int_{- 3}^{0} x^{2} d x + \int_{- 3}^{0} x^{3} d x$

Step 6

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

$- 12 (\frac{x^{2}}{2}]_{- 3}^{0}) - (\frac{1}{3} x^{3}]_{- 3}^{0}) + \int_{- 3}^{0} x^{3} d x$

Step 7

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

$- 12 (\frac{x^{2}}{2}]_{- 3}^{0}) - (\frac{x^{3}}{3}]_{- 3}^{0}) + \int_{- 3}^{0} x^{3} d x$

Step 8

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

$- 12 (\frac{x^{2}}{2}]_{- 3}^{0}) - (\frac{x^{3}}{3}]_{- 3}^{0}) + \frac{1}{4} x^{4}]_{- 3}^{0}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 3$ .

$- 12 ((\frac{0^{2}}{2}) - \frac{{(- 3)}^{2}}{2}) - (\frac{x^{3}}{3}]_{- 3}^{0}) + \frac{1}{4} x^{4}]_{- 3}^{0}$

Step 9.2

Evaluate $\frac{x^{3}}{3}$ at $0$ and at $- 3$ .

$- 12 (\frac{0^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + \frac{1}{4} x^{4}]_{- 3}^{0}$

Step 9.3

Evaluate $\frac{1}{4} x^{4}$ at $0$ and at $- 3$ .

$- 12 (\frac{0^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4

Simplify.

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Step 9.4.1

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

$- 12 (\frac{0}{2} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.2

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

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Step 9.4.2.1

Factor $2$ out of $0$ .

$- 12 (\frac{2 (0)}{2} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $2$ out of $2$ .

$- 12 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.2.2.2

Cancel the common factor.

$- 12 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.2.2.3

Rewrite the expression.

$- 12 (\frac{0}{1} - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.2.2.4

Divide $0$ by $1$ .

$- 12 (0 - \frac{{(- 3)}^{2}}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.3

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

$- 12 (0 - \frac{9}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.4

Subtract $\frac{9}{2}$ from $0$ .

$- 12 (- \frac{9}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.5

Multiply $- 1$ by $- 12$ .

$12 (\frac{9}{2}) - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.6

Combine $12$ and $\frac{9}{2}$ .

$\frac{12 \cdot 9}{2} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.7

Multiply $12$ by $9$ .

$\frac{108}{2} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.8

Cancel the common factor of $108$ and $2$ .

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Step 9.4.8.1

Factor $2$ out of $108$ .

$\frac{2 \cdot 54}{2} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.8.2

Cancel the common factors.

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Step 9.4.8.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 54}{2 (1)} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.8.2.2

Cancel the common factor.

$\frac{2 \cdot 54}{2 \cdot 1} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.8.2.3

Rewrite the expression.

$\frac{54}{1} - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.8.2.4

Divide $54$ by $1$ .

$54 - (\frac{0^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.9

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

$54 - (\frac{0}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.10

Cancel the common factor of $0$ and $3$ .

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Step 9.4.10.1

Factor $3$ out of $0$ .

$54 - (\frac{3 (0)}{3} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.10.2

Cancel the common factors.

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Step 9.4.10.2.1

Factor $3$ out of $3$ .

$54 - (\frac{3 \cdot 0}{3 \cdot 1} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.10.2.2

Cancel the common factor.

$54 - (\frac{3 \cdot 0}{3 \cdot 1} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.10.2.3

Rewrite the expression.

$54 - (\frac{0}{1} - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.10.2.4

Divide $0$ by $1$ .

$54 - (0 - \frac{{(- 3)}^{3}}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.11

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

$54 - (0 - \frac{- 27}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.12

Cancel the common factor of $- 27$ and $3$ .

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Step 9.4.12.1

Factor $3$ out of $- 27$ .

$54 - (0 - \frac{3 \cdot - 9}{3}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.12.2

Cancel the common factors.

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Step 9.4.12.2.1

Factor $3$ out of $3$ .

$54 - (0 - \frac{3 \cdot - 9}{3 (1)}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.12.2.2

Cancel the common factor.

$54 - (0 - \frac{3 \cdot - 9}{3 \cdot 1}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.12.2.3

Rewrite the expression.

$54 - (0 - \frac{- 9}{1}) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.12.2.4

Divide $- 9$ by $1$ .

$54 - (0 - - 9) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.13

Multiply $- 1$ by $- 9$ .

$54 - (0 + 9) + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.14

Add $0$ and $9$ .

$54 - 1 \cdot 9 + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.15

Multiply $- 1$ by $9$ .

$54 - 9 + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.16

Subtract $9$ from $54$ .

$45 + (\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.17

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

$45 + \frac{1}{4} \cdot 0 - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.18

Multiply $\frac{1}{4}$ by $0$ .

$45 + 0 - \frac{1}{4} {(- 3)}^{4}$

Step 9.4.19

Raise $- 3$ to the power of $4$ .

$45 + 0 - \frac{1}{4} \cdot 81$

Step 9.4.20

Multiply $81$ by $- 1$ .

$45 + 0 - 81 (\frac{1}{4})$

Step 9.4.21

Combine $- 81$ and $\frac{1}{4}$ .

$45 + 0 + \frac{- 81}{4}$

Step 9.4.22

Move the negative in front of the fraction.

$45 + 0 - \frac{81}{4}$

Step 9.4.23

Subtract $\frac{81}{4}$ from $0$ .

$45 - \frac{81}{4}$

Step 9.4.24

To write $45$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$45 \cdot \frac{4}{4} - \frac{81}{4}$

Step 9.4.25

Combine $45$ and $\frac{4}{4}$ .

$\frac{45 \cdot 4}{4} - \frac{81}{4}$

Step 9.4.26

Combine the numerators over the common denominator.

$\frac{45 \cdot 4 - 81}{4}$

Step 9.4.27

Simplify the numerator.

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Step 9.4.27.1

Multiply $45$ by $4$ .

$\frac{180 - 81}{4}$

Step 9.4.27.2

Subtract $81$ from $180$ .

$\frac{99}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{99}{4}$

Decimal Form:

$24.75$

Mixed Number Form:

$24 \frac{3}{4}$

Step 11