Calculus Examples

$\int_{0}^{3500} 150 - 0.04 x + 0.0002 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3500} 150 d x + \int_{0}^{3500} - 0.04 x d x + \int_{0}^{3500} 0.0002 x^{2} d x$

Step 2

Apply the constant rule.

$150 x]_{0}^{3500} + \int_{0}^{3500} - 0.04 x d x + \int_{0}^{3500} 0.0002 x^{2} d x$

Step 3

Since $- 0.04$ is constant with respect to $x$ , move $- 0.04$ out of the integral.

$150 x]_{0}^{3500} - 0.04 \int_{0}^{3500} x d x + \int_{0}^{3500} 0.0002 x^{2} d x$

Step 4

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

$150 x]_{0}^{3500} - 0.04 (\frac{1}{2} x^{2}]_{0}^{3500}) + \int_{0}^{3500} 0.0002 x^{2} d x$

Step 5

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

$150 x]_{0}^{3500} - 0.04 (\frac{x^{2}}{2}]_{0}^{3500}) + \int_{0}^{3500} 0.0002 x^{2} d x$

Step 6

Since $0.0002$ is constant with respect to $x$ , move $0.0002$ out of the integral.

$150 x]_{0}^{3500} - 0.04 (\frac{x^{2}}{2}]_{0}^{3500}) + 0.0002 \int_{0}^{3500} x^{2} d x$

Step 7

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

$150 x]_{0}^{3500} - 0.04 (\frac{x^{2}}{2}]_{0}^{3500}) + 0.0002 (\frac{1}{3} x^{3}]_{0}^{3500})$

Step 8

Simplify the answer.

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Step 8.1

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

$150 x]_{0}^{3500} - 0.04 (\frac{x^{2}}{2}]_{0}^{3500}) + 0.0002 (\frac{x^{3}}{3}]_{0}^{3500})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $150 x$ at $3500$ and at $0$ .

$(150 \cdot 3500) - 150 \cdot 0 - 0.04 (\frac{x^{2}}{2}]_{0}^{3500}) + 0.0002 (\frac{x^{3}}{3}]_{0}^{3500})$

Step 8.2.2

Evaluate $\frac{x^{2}}{2}$ at $3500$ and at $0$ .

$150 \cdot 3500 - 150 \cdot 0 - 0.04 (\frac{3500^{2}}{2} - \frac{0^{2}}{2}) + 0.0002 (\frac{x^{3}}{3}]_{0}^{3500})$

Step 8.2.3

Evaluate $\frac{x^{3}}{3}$ at $3500$ and at $0$ .

$150 \cdot 3500 - 150 \cdot 0 - 0.04 (\frac{3500^{2}}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4

Simplify.

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Step 8.2.4.1

Multiply $150$ by $3500$ .

$525000 - 150 \cdot 0 - 0.04 (\frac{3500^{2}}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.2

Multiply $- 150$ by $0$ .

$525000 + 0 - 0.04 (\frac{3500^{2}}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.3

Add $525000$ and $0$ .

$525000 - 0.04 (\frac{3500^{2}}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.4

Raise $3500$ to the power of $2$ .

$525000 - 0.04 (\frac{12250000}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5

Cancel the common factor of $12250000$ and $2$ .

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Step 8.2.4.5.1

Factor $2$ out of $12250000$ .

$525000 - 0.04 (\frac{2 \cdot 6125000}{2} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5.2

Cancel the common factors.

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Step 8.2.4.5.2.1

Factor $2$ out of $2$ .

$525000 - 0.04 (\frac{2 \cdot 6125000}{2 (1)} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5.2.2

Cancel the common factor.

$525000 - 0.04 (\frac{2 \cdot 6125000}{2 \cdot 1} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5.2.3

Rewrite the expression.

$525000 - 0.04 (\frac{6125000}{1} - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5.2.4

Divide $6125000$ by $1$ .

$525000 - 0.04 (6125000 - \frac{0^{2}}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6

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

$525000 - 0.04 (6125000 - \frac{0}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7

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

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Step 8.2.4.7.1

Factor $2$ out of $0$ .

$525000 - 0.04 (6125000 - \frac{2 (0)}{2}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7.2

Cancel the common factors.

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Step 8.2.4.7.2.1

Factor $2$ out of $2$ .

$525000 - 0.04 (6125000 - \frac{2 \cdot 0}{2 \cdot 1}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7.2.2

Cancel the common factor.

$525000 - 0.04 (6125000 - \frac{2 \cdot 0}{2 \cdot 1}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7.2.3

Rewrite the expression.

$525000 - 0.04 (6125000 - \frac{0}{1}) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7.2.4

Divide $0$ by $1$ .

$525000 - 0.04 (6125000 - 0) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.8

Multiply $- 1$ by $0$ .

$525000 - 0.04 (6125000 + 0) + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.9

Add $6125000$ and $0$ .

$525000 - 0.04 \cdot 6125000 + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.10

Multiply $- 0.04$ by $6125000$ .

$525000 - 245000 + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11

Subtract $245000$ from $525000$ .

$280000 + 0.0002 ((\frac{3500^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.12

Raise $3500$ to the power of $3$ .

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{0^{3}}{3})$

Step 8.2.4.13

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

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{0}{3})$

Step 8.2.4.14

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

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Step 8.2.4.14.1

Factor $3$ out of $0$ .

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{3 (0)}{3})$

Step 8.2.4.14.2

Cancel the common factors.

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Step 8.2.4.14.2.1

Factor $3$ out of $3$ .

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.4.14.2.2

Cancel the common factor.

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.4.14.2.3

Rewrite the expression.

$280000 + 0.0002 (\frac{42875000000}{3} - \frac{0}{1})$

Step 8.2.4.14.2.4

Divide $0$ by $1$ .

$280000 + 0.0002 (\frac{42875000000}{3} - 0)$

Step 8.2.4.15

Multiply $- 1$ by $0$ .

$280000 + 0.0002 (\frac{42875000000}{3} + 0)$

Step 8.2.4.16

Add $\frac{42875000000}{3}$ and $0$ .

$280000 + 0.0002 (\frac{42875000000}{3})$

Step 8.2.4.17

Combine $0.0002$ and $\frac{42875000000}{3}$ .

$280000 + \frac{0.0002 \cdot 42875000000}{3}$

Step 8.2.4.18

Multiply $0.0002$ by $42875000000$ .

$280000 + \frac{8575000}{3}$

Step 8.2.4.19

Cancel the common factor of $8575000$ and $3$ .

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Step 8.2.4.19.1

Rewrite $8575000$ as $1 (8575000)$ .

$280000 + \frac{1 (8575000)}{3}$

Step 8.2.4.19.2

Cancel the common factors.

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Step 8.2.4.19.2.1

Rewrite $3$ as $1 (3)$ .

$280000 + \frac{1 \cdot 8575000}{1 \cdot 3}$

Step 8.2.4.19.2.2

Cancel the common factor.

$280000 + \frac{1 \cdot 8575000}{1 \cdot 3}$

Step 8.2.4.19.2.3

Rewrite the expression.

$280000 + \frac{8575000}{3}$

Step 8.2.4.20

To write $280000$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$280000 \cdot \frac{3}{3} + \frac{8575000}{3}$

Step 8.2.4.21

Combine $280000$ and $\frac{3}{3}$ .

$\frac{280000 \cdot 3}{3} + \frac{8575000}{3}$

Step 8.2.4.22

Combine the numerators over the common denominator.

$\frac{280000 \cdot 3 + 8575000}{3}$

Step 8.2.4.23

Simplify the numerator.

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Step 8.2.4.23.1

Multiply $280000$ by $3$ .

$\frac{840000 + 8575000}{3}$

Step 8.2.4.23.2

Add $840000$ and $8575000$ .

$\frac{9415000}{3}$

Step 8.2.4.24

Cancel the common factor of $9415000$ and $3$ .

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Step 8.2.4.24.1

Rewrite $9415000$ as $1 (9415000)$ .

$\frac{1 (9415000)}{3}$

Step 8.2.4.24.2

Cancel the common factors.

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Step 8.2.4.24.2.1

Rewrite $3$ as $1 (3)$ .

$\frac{1 \cdot 9415000}{1 \cdot 3}$

Step 8.2.4.24.2.2

Cancel the common factor.

$\frac{1 \cdot 9415000}{1 \cdot 3}$

Step 8.2.4.24.2.3

Rewrite the expression.

$\frac{9415000}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{9415000}{3}$

Decimal Form:

$3138333.\overline{3}$

Mixed Number Form:

$3138333 \frac{1}{3}$

Step 10