Calculus Examples

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

Step 1

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

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

Step 2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{9} (x - 1) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 3.1

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

$\int_{1}^{9} (x - 1) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int_{1}^{9} (x - 1) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{9} (x - 1) x^{- \frac{1}{2}} d x$

Step 4

Expand $(x - 1) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

Raise $x$ to the power of $1$ .

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

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.4

Write $1$ as a fraction with a common denominator.

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Subtract $1$ from $2$ .

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

Step 5

Rewrite $- 1 x^{- \frac{1}{2}}$ as $- x^{- \frac{1}{2}}$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $9$ and at $1$ .

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

Step 10.2

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

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

Step 10.3

Simplify.

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Step 10.3.1

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

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

Step 10.3.2

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

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

Step 10.3.3

Cancel the common factor of $2$ .

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Step 10.3.3.1

Cancel the common factor.

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

Step 10.3.3.2

Rewrite the expression.

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

Step 10.3.4

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

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

Step 10.3.5

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

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

Step 10.3.6

Multiply $2$ by $27$ .

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

Step 10.3.7

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

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Step 10.3.7.1

Factor $3$ out of $54$ .

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

Step 10.3.7.2

Cancel the common factors.

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Step 10.3.7.2.1

Factor $3$ out of $3$ .

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

Step 10.3.7.2.2

Cancel the common factor.

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

Step 10.3.7.2.3

Rewrite the expression.

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

Step 10.3.7.2.4

Divide $18$ by $1$ .

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

Step 10.3.8

One to any power is one.

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

Step 10.3.9

Multiply $- 1$ by $1$ .

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

Step 10.3.10

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.3.11

Combine $18$ and $\frac{3}{3}$ .

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

Step 10.3.12

Combine the numerators over the common denominator.

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

Step 10.3.13

Simplify the numerator.

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Step 10.3.13.1

Multiply $18$ by $3$ .

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

Step 10.3.13.2

Subtract $2$ from $54$ .

$\frac{52}{3} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.14

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

$\frac{52}{3} - (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.15

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

$\frac{52}{3} - (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.16

Cancel the common factor of $2$ .

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Step 10.3.16.1

Cancel the common factor.

$\frac{52}{3} - (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.16.2

Rewrite the expression.

$\frac{52}{3} - (2 \cdot 3^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.17

Evaluate the exponent.

$\frac{52}{3} - (2 \cdot 3 - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.18

Multiply $2$ by $3$ .

$\frac{52}{3} - (6 - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.19

One to any power is one.

$\frac{52}{3} - (6 - 2 \cdot 1)$

Step 10.3.20

Multiply $- 2$ by $1$ .

$\frac{52}{3} - (6 - 2)$

Step 10.3.21

Subtract $2$ from $6$ .

$\frac{52}{3} - 1 \cdot 4$

Step 10.3.22

Multiply $- 1$ by $4$ .

$\frac{52}{3} - 4$

Step 10.3.23

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{52}{3} - 4 \cdot \frac{3}{3}$

Step 10.3.24

Combine $- 4$ and $\frac{3}{3}$ .

$\frac{52}{3} + \frac{- 4 \cdot 3}{3}$

Step 10.3.25

Combine the numerators over the common denominator.

$\frac{52 - 4 \cdot 3}{3}$

Step 10.3.26

Simplify the numerator.

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Step 10.3.26.1

Multiply $- 4$ by $3$ .

$\frac{52 - 12}{3}$

Step 10.3.26.2

Subtract $12$ from $52$ .

$\frac{40}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{3}$

Decimal Form:

$13.\overline{3}$

Mixed Number Form:

$13 \frac{1}{3}$

Step 12