Calculus Examples

$\int_{0}^{3} (3 - 2 x + x^{2}) d x = 9$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

$3 x]_{0}^{3} + \int_{0}^{3} - 2 x d x + \int_{0}^{3} x^{2} d x$

Step 4

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

$3 x]_{0}^{3} - 2 \int_{0}^{3} x d x + \int_{0}^{3} x^{2} d x$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

Multiply $3$ by $3$ .

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

Step 8.2.3.2

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

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

Step 8.2.3.3

Combine $\frac{1}{3}$ and $27$ .

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

Step 8.2.3.4

Cancel the common factor of $27$ and $3$ .

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Step 8.2.3.4.1

Factor $3$ out of $27$ .

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

Step 8.2.3.4.2

Cancel the common factors.

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Step 8.2.3.4.2.1

Factor $3$ out of $3$ .

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

Step 8.2.3.4.2.2

Cancel the common factor.

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

Step 8.2.3.4.2.3

Rewrite the expression.

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

Step 8.2.3.4.2.4

Divide $9$ by $1$ .

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

Step 8.2.3.5

Add $9$ and $9$ .

$18 - (3 \cdot 0 + \frac{1}{3} \cdot 0^{3}) - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.6

Multiply $3$ by $0$ .

$18 - (0 + \frac{1}{3} \cdot 0^{3}) - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7

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

$18 - (0 + \frac{1}{3} \cdot 0) - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.8

Multiply $\frac{1}{3}$ by $0$ .

$18 - (0 + 0) - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.9

Add $0$ and $0$ .

$18 - 0 - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.10

Multiply $- 1$ by $0$ .

$18 + 0 - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.11

Add $18$ and $0$ .

$18 - 2 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.12

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

$18 - 2 (\frac{9}{2} - \frac{0^{2}}{2})$

Step 8.2.3.13

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

$18 - 2 (\frac{9}{2} - \frac{0}{2})$

Step 8.2.3.14

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

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Step 8.2.3.14.1

Factor $2$ out of $0$ .

$18 - 2 (\frac{9}{2} - \frac{2 (0)}{2})$

Step 8.2.3.14.2

Cancel the common factors.

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Step 8.2.3.14.2.1

Factor $2$ out of $2$ .

$18 - 2 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.14.2.2

Cancel the common factor.

$18 - 2 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.14.2.3

Rewrite the expression.

$18 - 2 (\frac{9}{2} - \frac{0}{1})$

Step 8.2.3.14.2.4

Divide $0$ by $1$ .

$18 - 2 (\frac{9}{2} - 0)$

Step 8.2.3.15

Multiply $- 1$ by $0$ .

$18 - 2 (\frac{9}{2} + 0)$

Step 8.2.3.16

Add $\frac{9}{2}$ and $0$ .

$18 - 2 (\frac{9}{2})$

Step 8.2.3.17

Combine $- 2$ and $\frac{9}{2}$ .

$18 + \frac{- 2 \cdot 9}{2}$

Step 8.2.3.18

Multiply $- 2$ by $9$ .

$18 + \frac{- 18}{2}$

Step 8.2.3.19

Cancel the common factor of $- 18$ and $2$ .

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Step 8.2.3.19.1

Factor $2$ out of $- 18$ .

$18 + \frac{2 \cdot - 9}{2}$

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 $2$ .

$18 + \frac{2 \cdot - 9}{2 (1)}$

Step 8.2.3.19.2.2

Cancel the common factor.

$18 + \frac{2 \cdot - 9}{2 \cdot 1}$

Step 8.2.3.19.2.3

Rewrite the expression.

$18 + \frac{- 9}{1}$

Step 8.2.3.19.2.4

Divide $- 9$ by $1$ .

$18 - 9$

Step 8.2.3.20

Subtract $9$ from $18$ .

$9$

Step 9