Calculus Examples

$2 \int_{0}^{4.1} - 0.2 x^{2} (x - 5) d x$

Step 1

Multiply $- 0.2 x^{2} (x - 5)$ .

$2 \int_{0}^{4.1} - 0.2 x^{2} x - 0.2 x^{2} \cdot - 5 d x$

Step 2

Simplify.

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Step 2.1

Raise $x$ to the power of $1$ .

$2 \int_{0}^{4.1} - 0.2 (x^{1} x^{2}) - 0.2 x^{2} \cdot - 5 d x$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \int_{0}^{4.1} - 0.2 x^{1 + 2} - 0.2 x^{2} \cdot - 5 d x$

Step 2.3

Add $1$ and $2$ .

$2 \int_{0}^{4.1} - 0.2 x^{3} - 0.2 x^{2} \cdot - 5 d x$

Step 2.4

Multiply $- 5$ by $- 0.2$ .

$2 \int_{0}^{4.1} - 0.2 x^{3} + 1 x^{2} d x$

Step 2.5

Multiply $x^{2}$ by $1$ .

$2 \int_{0}^{4.1} - 0.2 x^{3} + x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$2 (\int_{0}^{4.1} - 0.2 x^{3} d x + \int_{0}^{4.1} x^{2} d x)$

Step 4

Since $- 0.2$ is constant with respect to $x$ , move $- 0.2$ out of the integral.

$2 (- 0.2 \int_{0}^{4.1} x^{3} d x + \int_{0}^{4.1} x^{2} d x)$

Step 5

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

$2 (- 0.2 (\frac{1}{4} x^{4}]_{0}^{4.1}) + \int_{0}^{4.1} x^{2} d x)$

Step 6

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

$2 (- 0.2 (\frac{x^{4}}{4}]_{0}^{4.1}) + \int_{0}^{4.1} x^{2} d x)$

Step 7

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

$2 (- 0.2 (\frac{x^{4}}{4}]_{0}^{4.1}) + \frac{1}{3} x^{3}]_{0}^{4.1})$

Step 8

Simplify the answer.

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Step 8.1

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

$2 (- 0.2 (\frac{x^{4}}{4}]_{0}^{4.1}) + \frac{x^{3}}{3}]_{0}^{4.1})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{4}}{4}$ at $4.1$ and at $0$ .

$2 (- 0.2 ((\frac{{4.1}^{4}}{4}) - \frac{0^{4}}{4}) + \frac{x^{3}}{3}]_{0}^{4.1})$

Step 8.2.2

Evaluate $\frac{x^{3}}{3}$ at $4.1$ and at $0$ .

$2 (- 0.2 (\frac{{4.1}^{4}}{4} - \frac{0^{4}}{4}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Raise $4.1$ to the power of $4$ .

$2 (- 0.2 (\frac{282.5761}{4} - \frac{0^{4}}{4}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.2

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

$2 (- 0.2 (\frac{282.5761}{4} - \frac{0}{4}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.3

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

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Step 8.2.3.3.1

Factor $4$ out of $0$ .

$2 (- 0.2 (\frac{282.5761}{4} - \frac{4 (0)}{4}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.3.2

Cancel the common factors.

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Step 8.2.3.3.2.1

Factor $4$ out of $4$ .

$2 (- 0.2 (\frac{282.5761}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.3.2.2

Cancel the common factor.

$2 (- 0.2 (\frac{282.5761}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.3.2.3

Rewrite the expression.

$2 (- 0.2 (\frac{282.5761}{4} - \frac{0}{1}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.3.2.4

Divide $0$ by $1$ .

$2 (- 0.2 (\frac{282.5761}{4} - 0) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.4

Multiply $- 1$ by $0$ .

$2 (- 0.2 (\frac{282.5761}{4} + 0) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.5

Add $\frac{282.5761}{4}$ and $0$ .

$2 (- 0.2 (\frac{282.5761}{4}) + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.6

Combine $- 0.2$ and $\frac{282.5761}{4}$ .

$2 (\frac{- 0.2 \cdot 282.5761}{4} + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.7

Multiply $- 0.2$ by $282.5761$ .

$2 (\frac{- 56.51522}{4} + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.8

Move the negative in front of the fraction.

$2 (- \frac{56.51522}{4} + \frac{{4.1}^{3}}{3} - \frac{0^{3}}{3})$

Step 8.2.3.9

Raise $4.1$ to the power of $3$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{0^{3}}{3})$

Step 8.2.3.10

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

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{0}{3})$

Step 8.2.3.11

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

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Step 8.2.3.11.1

Factor $3$ out of $0$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{3 (0)}{3})$

Step 8.2.3.11.2

Cancel the common factors.

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Step 8.2.3.11.2.1

Factor $3$ out of $3$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.11.2.2

Cancel the common factor.

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.11.2.3

Rewrite the expression.

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - \frac{0}{1})$

Step 8.2.3.11.2.4

Divide $0$ by $1$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} - 0)$

Step 8.2.3.12

Multiply $- 1$ by $0$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3} + 0)$

Step 8.2.3.13

Add $\frac{68.921}{3}$ and $0$ .

$2 (- \frac{56.51522}{4} + \frac{68.921}{3})$

Step 8.2.3.14

To write $- \frac{56.51522}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 (- \frac{56.51522}{4} \cdot \frac{3}{3} + \frac{68.921}{3})$

Step 8.2.3.15

To write $\frac{68.921}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (- \frac{56.51522}{4} \cdot \frac{3}{3} + \frac{68.921}{3} \cdot \frac{4}{4})$

Step 8.2.3.16

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.16.1

Multiply $\frac{56.51522}{4}$ by $\frac{3}{3}$ .

$2 (- \frac{56.51522 \cdot 3}{4 \cdot 3} + \frac{68.921}{3} \cdot \frac{4}{4})$

Step 8.2.3.16.2

Multiply $4$ by $3$ .

$2 (- \frac{56.51522 \cdot 3}{12} + \frac{68.921}{3} \cdot \frac{4}{4})$

Step 8.2.3.16.3

Multiply $\frac{68.921}{3}$ by $\frac{4}{4}$ .

$2 (- \frac{56.51522 \cdot 3}{12} + \frac{68.921 \cdot 4}{3 \cdot 4})$

Step 8.2.3.16.4

Multiply $3$ by $4$ .

$2 (- \frac{56.51522 \cdot 3}{12} + \frac{68.921 \cdot 4}{12})$

Step 8.2.3.17

Combine the numerators over the common denominator.

$2 \frac{- 56.51522 \cdot 3 + 68.921 \cdot 4}{12}$

Step 8.2.3.18

Multiply $- 56.51522$ by $3$ .

$2 \frac{- 169.54566 + 68.921 \cdot 4}{12}$

Step 8.2.3.19

Multiply $68.921$ by $4$ .

$2 \frac{- 169.54566 + 275.684}{12}$

Step 8.2.3.20

Add $- 169.54566$ and $275.684$ .

$2 (\frac{106.13834}{12})$

Step 8.2.3.21

Combine $2$ and $\frac{106.13834}{12}$ .

$\frac{2 \cdot 106.13834}{12}$

Step 8.2.3.22

Multiply $2$ by $106.13834$ .

$\frac{212.27668}{12}$

Step 9

Divide $212.27668$ by $12$ .

$17.68972\overline{3}$

Step 10