Calculus Examples

$\int_{0}^{300} 0.0003 x^{2} - 0.12 x + 20 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{300} 0.0003 x^{2} d x + \int_{0}^{300} - 0.12 x d x + \int_{0}^{300} 20 d x$

Step 2

Since $0.0003$ is constant with respect to $x$ , move $0.0003$ out of the integral.

$0.0003 \int_{0}^{300} x^{2} d x + \int_{0}^{300} - 0.12 x d x + \int_{0}^{300} 20 d x$

Step 3

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

$0.0003 (\frac{1}{3} x^{3}]_{0}^{300}) + \int_{0}^{300} - 0.12 x d x + \int_{0}^{300} 20 d x$

Step 4

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

$0.0003 (\frac{x^{3}}{3}]_{0}^{300}) + \int_{0}^{300} - 0.12 x d x + \int_{0}^{300} 20 d x$

Step 5

Since $- 0.12$ is constant with respect to $x$ , move $- 0.12$ out of the integral.

$0.0003 (\frac{x^{3}}{3}]_{0}^{300}) - 0.12 \int_{0}^{300} x d x + \int_{0}^{300} 20 d x$

Step 6

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

$0.0003 (\frac{x^{3}}{3}]_{0}^{300}) - 0.12 (\frac{1}{2} x^{2}]_{0}^{300}) + \int_{0}^{300} 20 d x$

Step 7

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

$0.0003 (\frac{x^{3}}{3}]_{0}^{300}) - 0.12 (\frac{x^{2}}{2}]_{0}^{300}) + \int_{0}^{300} 20 d x$

Step 8

Apply the constant rule.

$0.0003 (\frac{x^{3}}{3}]_{0}^{300}) - 0.12 (\frac{x^{2}}{2}]_{0}^{300}) + 20 x]_{0}^{300}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $300$ and at $0$ .

$0.0003 ((\frac{300^{3}}{3}) - \frac{0^{3}}{3}) - 0.12 (\frac{x^{2}}{2}]_{0}^{300}) + 20 x]_{0}^{300}$

Step 9.2

Evaluate $\frac{x^{2}}{2}$ at $300$ and at $0$ .

$0.0003 (\frac{300^{3}}{3} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + 20 x]_{0}^{300}$

Step 9.3

Evaluate $20 x$ at $300$ and at $0$ .

$0.0003 (\frac{300^{3}}{3} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4

Simplify.

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Step 9.4.1

Raise $300$ to the power of $3$ .

$0.0003 (\frac{27000000}{3} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.2

Cancel the common factor of $27000000$ and $3$ .

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Step 9.4.2.1

Factor $3$ out of $27000000$ .

$0.0003 (\frac{3 \cdot 9000000}{3} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $3$ out of $3$ .

$0.0003 (\frac{3 \cdot 9000000}{3 (1)} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.2.2.2

Cancel the common factor.

$0.0003 (\frac{3 \cdot 9000000}{3 \cdot 1} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.2.2.3

Rewrite the expression.

$0.0003 (\frac{9000000}{1} - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.2.2.4

Divide $9000000$ by $1$ .

$0.0003 (9000000 - \frac{0^{3}}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.3

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

$0.0003 (9000000 - \frac{0}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.4

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

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Step 9.4.4.1

Factor $3$ out of $0$ .

$0.0003 (9000000 - \frac{3 (0)}{3}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.4.2

Cancel the common factors.

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Step 9.4.4.2.1

Factor $3$ out of $3$ .

$0.0003 (9000000 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.4.2.2

Cancel the common factor.

$0.0003 (9000000 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.4.2.3

Rewrite the expression.

$0.0003 (9000000 - \frac{0}{1}) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.4.2.4

Divide $0$ by $1$ .

$0.0003 (9000000 - 0) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.5

Multiply $- 1$ by $0$ .

$0.0003 (9000000 + 0) - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.6

Add $9000000$ and $0$ .

$0.0003 \cdot 9000000 - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.7

Multiply $0.0003$ by $9000000$ .

$2700 - 0.12 (\frac{300^{2}}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.8

Raise $300$ to the power of $2$ .

$2700 - 0.12 (\frac{90000}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.9

Cancel the common factor of $90000$ and $2$ .

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Step 9.4.9.1

Factor $2$ out of $90000$ .

$2700 - 0.12 (\frac{2 \cdot 45000}{2} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.9.2

Cancel the common factors.

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Step 9.4.9.2.1

Factor $2$ out of $2$ .

$2700 - 0.12 (\frac{2 \cdot 45000}{2 (1)} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.9.2.2

Cancel the common factor.

$2700 - 0.12 (\frac{2 \cdot 45000}{2 \cdot 1} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.9.2.3

Rewrite the expression.

$2700 - 0.12 (\frac{45000}{1} - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.9.2.4

Divide $45000$ by $1$ .

$2700 - 0.12 (45000 - \frac{0^{2}}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.10

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

$2700 - 0.12 (45000 - \frac{0}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.11

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

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Step 9.4.11.1

Factor $2$ out of $0$ .

$2700 - 0.12 (45000 - \frac{2 (0)}{2}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.11.2

Cancel the common factors.

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Step 9.4.11.2.1

Factor $2$ out of $2$ .

$2700 - 0.12 (45000 - \frac{2 \cdot 0}{2 \cdot 1}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.11.2.2

Cancel the common factor.

$2700 - 0.12 (45000 - \frac{2 \cdot 0}{2 \cdot 1}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.11.2.3

Rewrite the expression.

$2700 - 0.12 (45000 - \frac{0}{1}) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.11.2.4

Divide $0$ by $1$ .

$2700 - 0.12 (45000 - 0) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.12

Multiply $- 1$ by $0$ .

$2700 - 0.12 (45000 + 0) + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.13

Add $45000$ and $0$ .

$2700 - 0.12 \cdot 45000 + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.14

Multiply $- 0.12$ by $45000$ .

$2700 - 5400 + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.15

Subtract $5400$ from $2700$ .

$- 2700 + (20 \cdot 300) - 20 \cdot 0$

Step 9.4.16

Multiply $20$ by $300$ .

$- 2700 + 6000 - 20 \cdot 0$

Step 9.4.17

Multiply $- 20$ by $0$ .

$- 2700 + 6000 + 0$

Step 9.4.18

Add $6000$ and $0$ .

$- 2700 + 6000$

Step 9.4.19

Add $- 2700$ and $6000$ .

$3300$

Step 10