Calculus Examples

$π \int_{0}^{1} (9 - 6 x + x^{2}) d x$

Step 1

Remove parentheses.

$π \int_{0}^{1} 9 - 6 x + x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$π (\int_{0}^{1} 9 d x + \int_{0}^{1} - 6 x d x + \int_{0}^{1} x^{2} d x)$

Step 3

Apply the constant rule.

$π (9 x]_{0}^{1} + \int_{0}^{1} - 6 x d x + \int_{0}^{1} x^{2} d x)$

Step 4

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$π (9 x]_{0}^{1} - 6 \int_{0}^{1} x d x + \int_{0}^{1} x^{2} d x)$

Step 5

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

$π (9 x]_{0}^{1} - 6 (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} x^{2} d x)$

Step 6

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

$π (9 x]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} x^{2} d x)$

Step 7

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

$π (9 x]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + \frac{1}{3} x^{3}]_{0}^{1})$

Step 8

Simplify the answer.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $9 x]_{0}^{1}$ and $\frac{1}{3} x^{3}]_{0}^{1}$ .

$π (9 x + \frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}))$

Step 8.1.2

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

$π (9 x + \frac{x^{3}}{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $9 x + \frac{x^{3}}{3}$ at $1$ and at $0$ .

$π ((9 \cdot 1 + \frac{1^{3}}{3}) - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 (\frac{x^{2}}{2}]_{0}^{1}))$

Step 8.2.2

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

$π ((9 \cdot 1 + \frac{1^{3}}{3}) - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Multiply $9$ by $1$ .

$π (9 + \frac{1^{3}}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.2

One to any power is one.

$π (9 + \frac{1}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.3

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π (9 \cdot \frac{3}{3} + \frac{1}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.4

Combine $9$ and $\frac{3}{3}$ .

$π (\frac{9 \cdot 3}{3} + \frac{1}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.5

Combine the numerators over the common denominator.

$π (\frac{9 \cdot 3 + 1}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.6

Simplify the numerator.

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Step 8.2.3.6.1

Multiply $9$ by $3$ .

$π (\frac{27 + 1}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.6.2

Add $27$ and $1$ .

$π (\frac{28}{3} - (9 \cdot 0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.7

Multiply $9$ by $0$ .

$π (\frac{28}{3} - (0 + \frac{0^{3}}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.8

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

$π (\frac{28}{3} - (0 + \frac{0}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.9

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

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Step 8.2.3.9.1

Factor $3$ out of $0$ .

$π (\frac{28}{3} - (0 + \frac{3 (0)}{3}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.9.2

Cancel the common factors.

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Step 8.2.3.9.2.1

Factor $3$ out of $3$ .

$π (\frac{28}{3} - (0 + \frac{3 \cdot 0}{3 \cdot 1}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.9.2.2

Cancel the common factor.

$π (\frac{28}{3} - (0 + \frac{3 \cdot 0}{3 \cdot 1}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.9.2.3

Rewrite the expression.

$π (\frac{28}{3} - (0 + \frac{0}{1}) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.9.2.4

Divide $0$ by $1$ .

$π (\frac{28}{3} - (0 + 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.10

Add $0$ and $0$ .

$π (\frac{28}{3} - 0 - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.11

Multiply $- 1$ by $0$ .

$π (\frac{28}{3} + 0 - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.12

Add $\frac{28}{3}$ and $0$ .

$π (\frac{28}{3} - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}))$

Step 8.2.3.13

One to any power is one.

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{0^{2}}{2}))$

Step 8.2.3.14

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

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{0}{2}))$

Step 8.2.3.15

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

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Step 8.2.3.15.1

Factor $2$ out of $0$ .

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{2 (0)}{2}))$

Step 8.2.3.15.2

Cancel the common factors.

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Step 8.2.3.15.2.1

Factor $2$ out of $2$ .

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 8.2.3.15.2.2

Cancel the common factor.

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 8.2.3.15.2.3

Rewrite the expression.

$π (\frac{28}{3} - 6 (\frac{1}{2} - \frac{0}{1}))$

Step 8.2.3.15.2.4

Divide $0$ by $1$ .

$π (\frac{28}{3} - 6 (\frac{1}{2} - 0))$

Step 8.2.3.16

Multiply $- 1$ by $0$ .

$π (\frac{28}{3} - 6 (\frac{1}{2} + 0))$

Step 8.2.3.17

Add $\frac{1}{2}$ and $0$ .

$π (\frac{28}{3} - 6 (\frac{1}{2}))$

Step 8.2.3.18

Combine $- 6$ and $\frac{1}{2}$ .

$π (\frac{28}{3} + \frac{- 6}{2})$

Step 8.2.3.19

Cancel the common factor of $- 6$ and $2$ .

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Step 8.2.3.19.1

Factor $2$ out of $- 6$ .

$π (\frac{28}{3} + \frac{2 \cdot - 3}{2})$

Step 8.2.3.19.2

Cancel the common factors.

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Step 8.2.3.19.2.1

Factor $2$ out of $2$ .

$π (\frac{28}{3} + \frac{2 \cdot - 3}{2 (1)})$

Step 8.2.3.19.2.2

Cancel the common factor.

$π (\frac{28}{3} + \frac{2 \cdot - 3}{2 \cdot 1})$

Step 8.2.3.19.2.3

Rewrite the expression.

$π (\frac{28}{3} + \frac{- 3}{1})$

Step 8.2.3.19.2.4

Divide $- 3$ by $1$ .

$π (\frac{28}{3} - 3)$

Step 8.2.3.20

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

$π (\frac{28}{3} - 3 \cdot \frac{3}{3})$

Step 8.2.3.21

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

$π (\frac{28}{3} + \frac{- 3 \cdot 3}{3})$

Step 8.2.3.22

Combine the numerators over the common denominator.

$π \frac{28 - 3 \cdot 3}{3}$

Step 8.2.3.23

Simplify the numerator.

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Step 8.2.3.23.1

Multiply $- 3$ by $3$ .

$π \frac{28 - 9}{3}$

Step 8.2.3.23.2

Subtract $9$ from $28$ .

$π \frac{19}{3}$

Step 8.2.3.24

Combine $π$ and $\frac{19}{3}$ .

$\frac{π \cdot 19}{3}$

Step 8.2.3.25

Move $19$ to the left of $π$ .

$\frac{19 π}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{19 π}{3}$

Decimal Form:

$19.89675347 \dots$

Step 10