Calculus Examples

$\int_{0}^{2} x^{3} - 5 x^{2} + 2 x + 8 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{2} x^{3} d x + \int_{0}^{2} - 5 x^{2} d x + \int_{0}^{2} 2 x d x + \int_{0}^{2} 8 d x$

Step 2

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

$\frac{1}{4} x^{4}]_{0}^{2} + \int_{0}^{2} - 5 x^{2} d x + \int_{0}^{2} 2 x d x + \int_{0}^{2} 8 d x$

Step 3

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{1}{4} x^{4}]_{0}^{2} - 5 \int_{0}^{2} x^{2} d x + \int_{0}^{2} 2 x d x + \int_{0}^{2} 8 d x$

Step 4

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

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{1}{3} x^{3}]_{0}^{2}) + \int_{0}^{2} 2 x d x + \int_{0}^{2} 8 d x$

Step 5

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

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + \int_{0}^{2} 2 x d x + \int_{0}^{2} 8 d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 \int_{0}^{2} x d x + \int_{0}^{2} 8 d x$

Step 7

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

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 (\frac{1}{2} x^{2}]_{0}^{2}) + \int_{0}^{2} 8 d x$

Step 8

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

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 (\frac{x^{2}}{2}]_{0}^{2}) + \int_{0}^{2} 8 d x$

Step 9

Apply the constant rule.

$\frac{1}{4} x^{4}]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 (\frac{x^{2}}{2}]_{0}^{2}) + 8 x]_{0}^{2}$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{4} x^{4}]_{0}^{2}$ and $8 x]_{0}^{2}$ .

$\frac{1}{4} x^{4} + 8 x]_{0}^{2} - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 (\frac{x^{2}}{2}]_{0}^{2})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{1}{4} x^{4} + 8 x$ at $2$ and at $0$ .

$(\frac{1}{4} \cdot 2^{4} + 8 \cdot 2) - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{x^{3}}{3}]_{0}^{2}) + 2 (\frac{x^{2}}{2}]_{0}^{2})$

Step 10.2.2

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

$(\frac{1}{4} \cdot 2^{4} + 8 \cdot 2) - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3}) + 2 (\frac{x^{2}}{2}]_{0}^{2})$

Step 10.2.3

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

$(\frac{1}{4} \cdot 2^{4} + 8 \cdot 2) - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4

Simplify.

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Step 10.2.4.1

Raise $2$ to the power of $4$ .

$\frac{1}{4} \cdot 16 + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.2

Combine $\frac{1}{4}$ and $16$ .

$\frac{16}{4} + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.3

Cancel the common factor of $16$ and $4$ .

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Step 10.2.4.3.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 4}{4} + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.3.2

Cancel the common factors.

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Step 10.2.4.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 4}{4 (1)} + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.3.2.2

Cancel the common factor.

$\frac{4 \cdot 4}{4 \cdot 1} + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.3.2.3

Rewrite the expression.

$\frac{4}{1} + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.3.2.4

Divide $4$ by $1$ .

$4 + 8 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.4

Multiply $8$ by $2$ .

$4 + 16 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.5

Add $4$ and $16$ .

$20 - (\frac{1}{4} \cdot 0^{4} + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.6

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

$20 - (\frac{1}{4} \cdot 0 + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.7

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

$20 - (0 + 8 \cdot 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.8

Multiply $8$ by $0$ .

$20 - (0 + 0) - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.9

Add $0$ and $0$ .

$20 - 0 - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.10

Multiply $- 1$ by $0$ .

$20 + 0 - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.11

Add $20$ and $0$ .

$20 - 5 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.12

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

$20 - 5 (\frac{8}{3} - \frac{0^{3}}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.13

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

$20 - 5 (\frac{8}{3} - \frac{0}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.14

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

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Step 10.2.4.14.1

Factor $3$ out of $0$ .

$20 - 5 (\frac{8}{3} - \frac{3 (0)}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.14.2

Cancel the common factors.

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Step 10.2.4.14.2.1

Factor $3$ out of $3$ .

$20 - 5 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.14.2.2

Cancel the common factor.

$20 - 5 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.14.2.3

Rewrite the expression.

$20 - 5 (\frac{8}{3} - \frac{0}{1}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.14.2.4

Divide $0$ by $1$ .

$20 - 5 (\frac{8}{3} - 0) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.15

Multiply $- 1$ by $0$ .

$20 - 5 (\frac{8}{3} + 0) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.16

Add $\frac{8}{3}$ and $0$ .

$20 - 5 (\frac{8}{3}) + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.17

Combine $- 5$ and $\frac{8}{3}$ .

$20 + \frac{- 5 \cdot 8}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.18

Multiply $- 5$ by $8$ .

$20 + \frac{- 40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.19

Move the negative in front of the fraction.

$20 - \frac{40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.20

To write $20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20 \cdot \frac{3}{3} - \frac{40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.21

Combine $20$ and $\frac{3}{3}$ .

$\frac{20 \cdot 3}{3} - \frac{40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.22

Combine the numerators over the common denominator.

$\frac{20 \cdot 3 - 40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.23

Simplify the numerator.

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Step 10.2.4.23.1

Multiply $20$ by $3$ .

$\frac{60 - 40}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.23.2

Subtract $40$ from $60$ .

$\frac{20}{3} + 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2})$

Step 10.2.4.24

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

$\frac{20}{3} + 2 (\frac{4}{2} - \frac{0^{2}}{2})$

Step 10.2.4.25

Cancel the common factor of $4$ and $2$ .

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Step 10.2.4.25.1

Factor $2$ out of $4$ .

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

Step 10.2.4.25.2

Cancel the common factors.

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Step 10.2.4.25.2.1

Factor $2$ out of $2$ .

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

Step 10.2.4.25.2.2

Cancel the common factor.

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

Step 10.2.4.25.2.3

Rewrite the expression.

$\frac{20}{3} + 2 (\frac{2}{1} - \frac{0^{2}}{2})$

Step 10.2.4.25.2.4

Divide $2$ by $1$ .

$\frac{20}{3} + 2 (2 - \frac{0^{2}}{2})$

Step 10.2.4.26

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

$\frac{20}{3} + 2 (2 - \frac{0}{2})$

Step 10.2.4.27

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

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Step 10.2.4.27.1

Factor $2$ out of $0$ .

$\frac{20}{3} + 2 (2 - \frac{2 (0)}{2})$

Step 10.2.4.27.2

Cancel the common factors.

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Step 10.2.4.27.2.1

Factor $2$ out of $2$ .

$\frac{20}{3} + 2 (2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 10.2.4.27.2.2

Cancel the common factor.

$\frac{20}{3} + 2 (2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 10.2.4.27.2.3

Rewrite the expression.

$\frac{20}{3} + 2 (2 - \frac{0}{1})$

Step 10.2.4.27.2.4

Divide $0$ by $1$ .

$\frac{20}{3} + 2 (2 - 0)$

Step 10.2.4.28

Multiply $- 1$ by $0$ .

$\frac{20}{3} + 2 (2 + 0)$

Step 10.2.4.29

Add $2$ and $0$ .

$\frac{20}{3} + 2 \cdot 2$

Step 10.2.4.30

Multiply $2$ by $2$ .

$\frac{20}{3} + 4$

Step 10.2.4.31

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

$\frac{20}{3} + 4 \cdot \frac{3}{3}$

Step 10.2.4.32

Combine $4$ and $\frac{3}{3}$ .

$\frac{20}{3} + \frac{4 \cdot 3}{3}$

Step 10.2.4.33

Combine the numerators over the common denominator.

$\frac{20 + 4 \cdot 3}{3}$

Step 10.2.4.34

Simplify the numerator.

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Step 10.2.4.34.1

Multiply $4$ by $3$ .

$\frac{20 + 12}{3}$

Step 10.2.4.34.2

Add $20$ and $12$ .

$\frac{32}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3}$

Decimal Form:

$10.\overline{6}$

Mixed Number Form:

$10 \frac{2}{3}$

Step 12