Calculus Examples

$\int_{0}^{9} (\sqrt{x} - \frac{1}{3} x) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $9$ and at $0$ .

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 7.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.3.3

Cancel the common factor of $2$ .

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Step 7.3.3.1

Cancel the common factor.

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

Step 7.3.3.2

Rewrite the expression.

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

Step 7.3.4

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

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

Step 7.3.5

Combine $\frac{2}{3}$ and $27$ .

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

Step 7.3.6

Multiply $2$ by $27$ .

$\frac{54}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.7

Cancel the common factor of $54$ and $3$ .

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Step 7.3.7.1

Factor $3$ out of $54$ .

$\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.7.2

Cancel the common factors.

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Step 7.3.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.7.2.2

Cancel the common factor.

$\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.7.2.3

Rewrite the expression.

$\frac{18}{1} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.7.2.4

Divide $18$ by $1$ .

$18 - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.8

Rewrite $0$ as $0^{2}$ .

$18 - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.9

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$18 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.10

Cancel the common factor of $2$ .

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Step 7.3.10.1

Cancel the common factor.

$18 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.10.2

Rewrite the expression.

$18 - \frac{2}{3} \cdot 0^{3} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.11

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

$18 - \frac{2}{3} \cdot 0 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.12

Multiply $0$ by $- 1$ .

$18 + 0 (\frac{2}{3}) - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.13

Multiply $0$ by $\frac{2}{3}$ .

$18 + 0 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.14

Add $18$ and $0$ .

$18 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.15

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

$18 - \frac{1}{3} (\frac{1}{2} \cdot 81 - \frac{1}{2} \cdot 0^{2})$

Step 7.3.16

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

$18 - \frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0^{2})$

Step 7.3.17

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

$18 - \frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0)$

Step 7.3.18

Multiply $0$ by $- 1$ .

$18 - \frac{1}{3} (\frac{81}{2} + 0 (\frac{1}{2}))$

Step 7.3.19

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

$18 - \frac{1}{3} (\frac{81}{2} + 0)$

Step 7.3.20

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

$18 - \frac{1}{3} \cdot \frac{81}{2}$

Step 7.3.21

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

$18 - \frac{81}{2 \cdot 3}$

Step 7.3.22

Multiply $2$ by $3$ .

$18 - \frac{81}{6}$

Step 7.3.23

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

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Step 7.3.23.1

Factor $3$ out of $81$ .

$18 - \frac{3 (27)}{6}$

Step 7.3.23.2

Cancel the common factors.

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Step 7.3.23.2.1

Factor $3$ out of $6$ .

$18 - \frac{3 \cdot 27}{3 \cdot 2}$

Step 7.3.23.2.2

Cancel the common factor.

$18 - \frac{3 \cdot 27}{3 \cdot 2}$

Step 7.3.23.2.3

Rewrite the expression.

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

Step 7.3.24

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

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

Step 7.3.25

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

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

Step 7.3.26

Combine the numerators over the common denominator.

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

Step 7.3.27

Simplify the numerator.

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Step 7.3.27.1

Multiply $18$ by $2$ .

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

Step 7.3.27.2

Subtract $27$ from $36$ .

$\frac{9}{2}$

Step 8

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 9