Calculus Examples

$\int_{0}^{9} \frac{1}{3} x - 2 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{9} \frac{1}{3} x d x + \int_{0}^{9} - 2 d x$

Step 2

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{0}^{9} x d x + \int_{0}^{9} - 2 d x$

Step 3

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

$\frac{1}{3} \frac{1}{2} x^{2}]_{0}^{9} + \int_{0}^{9} - 2 d x$

Step 4

Apply the constant rule.

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

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{2} x^{2}$ at $9$ and at $0$ .

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

Step 5.2

Evaluate $- 2 x$ at $9$ and at $0$ .

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

Step 5.3

Simplify.

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Step 5.3.1

Raise $9$ to the power of $2$ .

$\frac{1}{3} (\frac{1}{2} \cdot 81 - \frac{1}{2} \cdot 0^{2}) - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.2

Combine $\frac{1}{2}$ and $81$ .

$\frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0^{2}) - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.3

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

$\frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0) - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.4

Multiply $0$ by $- 1$ .

$\frac{1}{3} (\frac{81}{2} + 0 (\frac{1}{2})) - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.5

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{3} (\frac{81}{2} + 0) - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.6

Add $\frac{81}{2}$ and $0$ .

$\frac{1}{3} \cdot \frac{81}{2} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.7

Multiply $\frac{1}{3}$ by $\frac{81}{2}$ .

$\frac{81}{3 \cdot 2} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.8

Multiply $3$ by $2$ .

$\frac{81}{6} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.9

Cancel the common factor of $81$ and $6$ .

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Step 5.3.9.1

Factor $3$ out of $81$ .

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

Step 5.3.9.2

Cancel the common factors.

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Step 5.3.9.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 27}{3 \cdot 2} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.9.2.2

Cancel the common factor.

$\frac{3 \cdot 27}{3 \cdot 2} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.9.2.3

Rewrite the expression.

$\frac{27}{2} - 2 \cdot 9 + 2 \cdot 0$

Step 5.3.10

Multiply $- 2$ by $9$ .

$\frac{27}{2} - 18 + 2 \cdot 0$

Step 5.3.11

Multiply $2$ by $0$ .

$\frac{27}{2} - 18 + 0$

Step 5.3.12

Add $- 18$ and $0$ .

$\frac{27}{2} - 18$

Step 5.3.13

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27}{2} - 18 \cdot \frac{2}{2}$

Step 5.3.14

Combine $- 18$ and $\frac{2}{2}$ .

$\frac{27}{2} + \frac{- 18 \cdot 2}{2}$

Step 5.3.15

Combine the numerators over the common denominator.

$\frac{27 - 18 \cdot 2}{2}$

Step 5.3.16

Simplify the numerator.

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Step 5.3.16.1

Multiply $- 18$ by $2$ .

$\frac{27 - 36}{2}$

Step 5.3.16.2

Subtract $36$ from $27$ .

$\frac{- 9}{2}$

Step 5.3.17

Move the negative in front of the fraction.

$- \frac{9}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{9}{2}$

Decimal Form:

$- 4.5$

Mixed Number Form:

$- 4 \frac{1}{2}$

Step 7