Calculus Examples

$\int_{0}^{60} - 0.0009 x^{3} + 0.0782 x^{2} - 0.2079 x + 0.0952 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{60} - 0.0009 x^{3} d x + \int_{0}^{60} 0.0782 x^{2} d x + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 2

Since $- 0.0009$ is constant with respect to $x$ , move $- 0.0009$ out of the integral.

$- 0.0009 \int_{0}^{60} x^{3} d x + \int_{0}^{60} 0.0782 x^{2} d x + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 3

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

$- 0.0009 (\frac{1}{4} x^{4}]_{0}^{60}) + \int_{0}^{60} 0.0782 x^{2} d x + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 4

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

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + \int_{0}^{60} 0.0782 x^{2} d x + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 5

Since $0.0782$ is constant with respect to $x$ , move $0.0782$ out of the integral.

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 \int_{0}^{60} x^{2} d x + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 6

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

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{1}{3} x^{3}]_{0}^{60}) + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 7

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

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) + \int_{0}^{60} - 0.2079 x d x + \int_{0}^{60} 0.0952 d x$

Step 8

Since $- 0.2079$ is constant with respect to $x$ , move $- 0.2079$ out of the integral.

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) - 0.2079 \int_{0}^{60} x d x + \int_{0}^{60} 0.0952 d x$

Step 9

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

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) - 0.2079 (\frac{1}{2} x^{2}]_{0}^{60}) + \int_{0}^{60} 0.0952 d x$

Step 10

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

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) - 0.2079 (\frac{x^{2}}{2}]_{0}^{60}) + \int_{0}^{60} 0.0952 d x$

Step 11

Apply the constant rule.

$- 0.0009 (\frac{x^{4}}{4}]_{0}^{60}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) - 0.2079 (\frac{x^{2}}{2}]_{0}^{60}) + 0.0952 x]_{0}^{60}$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $\frac{x^{4}}{4}$ at $60$ and at $0$ .

$- 0.0009 ((\frac{60^{4}}{4}) - \frac{0^{4}}{4}) + 0.0782 (\frac{x^{3}}{3}]_{0}^{60}) - 0.2079 (\frac{x^{2}}{2}]_{0}^{60}) + 0.0952 x]_{0}^{60}$

Step 12.2

Evaluate $\frac{x^{3}}{3}$ at $60$ and at $0$ .

$- 0.0009 (\frac{60^{4}}{4} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{x^{2}}{2}]_{0}^{60}) + 0.0952 x]_{0}^{60}$

Step 12.3

Evaluate $\frac{x^{2}}{2}$ at $60$ and at $0$ .

$- 0.0009 (\frac{60^{4}}{4} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 ((\frac{60^{2}}{2}) - \frac{0^{2}}{2}) + 0.0952 x]_{0}^{60}$

Step 12.4

Evaluate $0.0952 x$ at $60$ and at $0$ .

$- 0.0009 (\frac{60^{4}}{4} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5

Simplify.

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Step 12.5.1

Raise $60$ to the power of $4$ .

$- 0.0009 (\frac{12960000}{4} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.2

Cancel the common factor of $12960000$ and $4$ .

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Step 12.5.2.1

Factor $4$ out of $12960000$ .

$- 0.0009 (\frac{4 \cdot 3240000}{4} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.2.2

Cancel the common factors.

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Step 12.5.2.2.1

Factor $4$ out of $4$ .

$- 0.0009 (\frac{4 \cdot 3240000}{4 (1)} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.2.2.2

Cancel the common factor.

$- 0.0009 (\frac{4 \cdot 3240000}{4 \cdot 1} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.2.2.3

Rewrite the expression.

$- 0.0009 (\frac{3240000}{1} - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.2.2.4

Divide $3240000$ by $1$ .

$- 0.0009 (3240000 - \frac{0^{4}}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.3

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

$- 0.0009 (3240000 - \frac{0}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.4

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

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Step 12.5.4.1

Factor $4$ out of $0$ .

$- 0.0009 (3240000 - \frac{4 (0)}{4}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.4.2

Cancel the common factors.

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Step 12.5.4.2.1

Factor $4$ out of $4$ .

$- 0.0009 (3240000 - \frac{4 \cdot 0}{4 \cdot 1}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.4.2.2

Cancel the common factor.

$- 0.0009 (3240000 - \frac{4 \cdot 0}{4 \cdot 1}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.4.2.3

Rewrite the expression.

$- 0.0009 (3240000 - \frac{0}{1}) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.4.2.4

Divide $0$ by $1$ .

$- 0.0009 (3240000 - 0) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.5

Multiply $- 1$ by $0$ .

$- 0.0009 (3240000 + 0) + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.6

Add $3240000$ and $0$ .

$- 0.0009 \cdot 3240000 + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.7

Multiply $- 0.0009$ by $3240000$ .

$- 2916 + 0.0782 (\frac{60^{3}}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.8

Raise $60$ to the power of $3$ .

$- 2916 + 0.0782 (\frac{216000}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.9

Cancel the common factor of $216000$ and $3$ .

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Step 12.5.9.1

Factor $3$ out of $216000$ .

$- 2916 + 0.0782 (\frac{3 \cdot 72000}{3} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.9.2

Cancel the common factors.

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Step 12.5.9.2.1

Factor $3$ out of $3$ .

$- 2916 + 0.0782 (\frac{3 \cdot 72000}{3 (1)} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.9.2.2

Cancel the common factor.

$- 2916 + 0.0782 (\frac{3 \cdot 72000}{3 \cdot 1} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.9.2.3

Rewrite the expression.

$- 2916 + 0.0782 (\frac{72000}{1} - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.9.2.4

Divide $72000$ by $1$ .

$- 2916 + 0.0782 (72000 - \frac{0^{3}}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.10

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

$- 2916 + 0.0782 (72000 - \frac{0}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.11

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

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Step 12.5.11.1

Factor $3$ out of $0$ .

$- 2916 + 0.0782 (72000 - \frac{3 (0)}{3}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.11.2

Cancel the common factors.

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Step 12.5.11.2.1

Factor $3$ out of $3$ .

$- 2916 + 0.0782 (72000 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.11.2.2

Cancel the common factor.

$- 2916 + 0.0782 (72000 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.11.2.3

Rewrite the expression.

$- 2916 + 0.0782 (72000 - \frac{0}{1}) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.11.2.4

Divide $0$ by $1$ .

$- 2916 + 0.0782 (72000 - 0) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.12

Multiply $- 1$ by $0$ .

$- 2916 + 0.0782 (72000 + 0) - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.13

Add $72000$ and $0$ .

$- 2916 + 0.0782 \cdot 72000 - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.14

Multiply $0.0782$ by $72000$ .

$- 2916 + 5630.4 - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.15

Add $- 2916$ and $5630.4$ .

$2714.4 - 0.2079 (\frac{60^{2}}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.16

Raise $60$ to the power of $2$ .

$2714.4 - 0.2079 (\frac{3600}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.17

Cancel the common factor of $3600$ and $2$ .

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Step 12.5.17.1

Factor $2$ out of $3600$ .

$2714.4 - 0.2079 (\frac{2 \cdot 1800}{2} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.17.2

Cancel the common factors.

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Step 12.5.17.2.1

Factor $2$ out of $2$ .

$2714.4 - 0.2079 (\frac{2 \cdot 1800}{2 (1)} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.17.2.2

Cancel the common factor.

$2714.4 - 0.2079 (\frac{2 \cdot 1800}{2 \cdot 1} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.17.2.3

Rewrite the expression.

$2714.4 - 0.2079 (\frac{1800}{1} - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.17.2.4

Divide $1800$ by $1$ .

$2714.4 - 0.2079 (1800 - \frac{0^{2}}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.18

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

$2714.4 - 0.2079 (1800 - \frac{0}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.19

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

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Step 12.5.19.1

Factor $2$ out of $0$ .

$2714.4 - 0.2079 (1800 - \frac{2 (0)}{2}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.19.2

Cancel the common factors.

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Step 12.5.19.2.1

Factor $2$ out of $2$ .

$2714.4 - 0.2079 (1800 - \frac{2 \cdot 0}{2 \cdot 1}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.19.2.2

Cancel the common factor.

$2714.4 - 0.2079 (1800 - \frac{2 \cdot 0}{2 \cdot 1}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.19.2.3

Rewrite the expression.

$2714.4 - 0.2079 (1800 - \frac{0}{1}) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.19.2.4

Divide $0$ by $1$ .

$2714.4 - 0.2079 (1800 - 0) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.20

Multiply $- 1$ by $0$ .

$2714.4 - 0.2079 (1800 + 0) + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.21

Add $1800$ and $0$ .

$2714.4 - 0.2079 \cdot 1800 + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.22

Multiply $- 0.2079$ by $1800$ .

$2714.4 - 374.22 + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.23

Subtract $374.22$ from $2714.4$ .

$2340.18 + 0.0952 \cdot 60 - 0.0952 \cdot 0$

Step 12.5.24

Multiply $0.0952$ by $60$ .

$2340.18 + 5.712 - 0.0952 \cdot 0$

Step 12.5.25

Multiply $- 0.0952$ by $0$ .

$2340.18 + 5.712 + 0$

Step 12.5.26

Add $5.712$ and $0$ .

$2340.18 + 5.712$

Step 12.5.27

Add $2340.18$ and $5.712$ .

$2345.892$

Step 13