Calculus Examples

$\int_{2}^{3} (3 - x) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $3 x$ at $3$ and at $2$ .

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

Step 6.2.2

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

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Multiply $3$ by $3$ .

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

Step 6.2.3.2

Multiply $- 3$ by $2$ .

$9 - 6 - (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 6.2.3.3

Subtract $6$ from $9$ .

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

Step 6.2.3.4

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

$3 - (\frac{9}{2} - \frac{2^{2}}{2})$

Step 6.2.3.5

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

$3 - (\frac{9}{2} - \frac{4}{2})$

Step 6.2.3.6

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

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Step 6.2.3.6.1

Factor $2$ out of $4$ .

$3 - (\frac{9}{2} - \frac{2 \cdot 2}{2})$

Step 6.2.3.6.2

Cancel the common factors.

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Step 6.2.3.6.2.1

Factor $2$ out of $2$ .

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

Step 6.2.3.6.2.2

Cancel the common factor.

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

Step 6.2.3.6.2.3

Rewrite the expression.

$3 - (\frac{9}{2} - \frac{2}{1})$

Step 6.2.3.6.2.4

Divide $2$ by $1$ .

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

Step 6.2.3.7

Multiply $- 1$ by $2$ .

$3 - (\frac{9}{2} - 2)$

Step 6.2.3.8

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 - (\frac{9}{2} - 2 \cdot \frac{2}{2})$

Step 6.2.3.9

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

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

Step 6.2.3.10

Combine the numerators over the common denominator.

$3 - \frac{9 - 2 \cdot 2}{2}$

Step 6.2.3.11

Simplify the numerator.

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Step 6.2.3.11.1

Multiply $- 2$ by $2$ .

$3 - \frac{9 - 4}{2}$

Step 6.2.3.11.2

Subtract $4$ from $9$ .

$3 - \frac{5}{2}$

Step 6.2.3.12

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 \cdot \frac{2}{2} - \frac{5}{2}$

Step 6.2.3.13

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 \cdot 2}{2} - \frac{5}{2}$

Step 6.2.3.14

Combine the numerators over the common denominator.

$\frac{3 \cdot 2 - 5}{2}$

Step 6.2.3.15

Simplify the numerator.

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Step 6.2.3.15.1

Multiply $3$ by $2$ .

$\frac{6 - 5}{2}$

Step 6.2.3.15.2

Subtract $5$ from $6$ .

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 8