Calculus Examples

$\int_{9}^{16} \sqrt{x} - 4 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{9}^{16} \sqrt{x} d x + \int_{9}^{16} - 4 d x$

Step 2

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

$\int_{9}^{16} x^{\frac{1}{2}} d x + \int_{9}^{16} - 4 d x$

Step 3

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

$\frac{2}{3} x^{\frac{3}{2}}]_{9}^{16} + \int_{9}^{16} - 4 d x$

Step 4

Apply the constant rule.

$\frac{2}{3} x^{\frac{3}{2}}]_{9}^{16} + - 4 x]_{9}^{16}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{2}{3} x^{\frac{3}{2}}]_{9}^{16}$ and $- 4 x]_{9}^{16}$ .

$\frac{2}{3} x^{\frac{3}{2}} - 4 x]_{9}^{16}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}} - 4 x$ at $16$ and at $9$ .

$(\frac{2}{3} \cdot 16^{\frac{3}{2}} - 4 \cdot 16) - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{2}{3} \cdot {(4^{2})}^{\frac{3}{2}} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.2

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

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.3

Cancel the common factor of $2$ .

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 4^{3} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.4

Raise $4$ to the power of $3$ .

$\frac{2}{3} \cdot 64 - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.5

Combine $\frac{2}{3}$ and $64$ .

$\frac{2 \cdot 64}{3} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.6

Multiply $2$ by $64$ .

$\frac{128}{3} - 4 \cdot 16 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.7

Multiply $- 4$ by $16$ .

$\frac{128}{3} - 64 - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.8

To write $- 64$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{128}{3} - 64 \cdot \frac{3}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.9

Combine $- 64$ and $\frac{3}{3}$ .

$\frac{128}{3} + \frac{- 64 \cdot 3}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.10

Combine the numerators over the common denominator.

$\frac{128 - 64 \cdot 3}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.11

Simplify the numerator.

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Step 5.2.2.11.1

Multiply $- 64$ by $3$ .

$\frac{128 - 192}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.11.2

Subtract $192$ from $128$ .

$\frac{- 64}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.12

Move the negative in front of the fraction.

$- \frac{64}{3} - (\frac{2}{3} \cdot 9^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.13

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

$- \frac{64}{3} - (\frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - 4 \cdot 9)$

Step 5.2.2.14

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

$- \frac{64}{3} - (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - 4 \cdot 9)$

Step 5.2.2.15

Cancel the common factor of $2$ .

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Step 5.2.2.15.1

Cancel the common factor.

$- \frac{64}{3} - (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - 4 \cdot 9)$

Step 5.2.2.15.2

Rewrite the expression.

$- \frac{64}{3} - (\frac{2}{3} \cdot 3^{3} - 4 \cdot 9)$

Step 5.2.2.16

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

$- \frac{64}{3} - (\frac{2}{3} \cdot 27 - 4 \cdot 9)$

Step 5.2.2.17

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

$- \frac{64}{3} - (\frac{2 \cdot 27}{3} - 4 \cdot 9)$

Step 5.2.2.18

Multiply $2$ by $27$ .

$- \frac{64}{3} - (\frac{54}{3} - 4 \cdot 9)$

Step 5.2.2.19

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

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Step 5.2.2.19.1

Factor $3$ out of $54$ .

$- \frac{64}{3} - (\frac{3 \cdot 18}{3} - 4 \cdot 9)$

Step 5.2.2.19.2

Cancel the common factors.

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Step 5.2.2.19.2.1

Factor $3$ out of $3$ .

$- \frac{64}{3} - (\frac{3 \cdot 18}{3 (1)} - 4 \cdot 9)$

Step 5.2.2.19.2.2

Cancel the common factor.

$- \frac{64}{3} - (\frac{3 \cdot 18}{3 \cdot 1} - 4 \cdot 9)$

Step 5.2.2.19.2.3

Rewrite the expression.

$- \frac{64}{3} - (\frac{18}{1} - 4 \cdot 9)$

Step 5.2.2.19.2.4

Divide $18$ by $1$ .

$- \frac{64}{3} - (18 - 4 \cdot 9)$

Step 5.2.2.20

Multiply $- 4$ by $9$ .

$- \frac{64}{3} - (18 - 36)$

Step 5.2.2.21

Subtract $36$ from $18$ .

$- \frac{64}{3} - - 18$

Step 5.2.2.22

Multiply $- 1$ by $- 18$ .

$- \frac{64}{3} + 18$

Step 5.2.2.23

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

$- \frac{64}{3} + 18 \cdot \frac{3}{3}$

Step 5.2.2.24

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

$- \frac{64}{3} + \frac{18 \cdot 3}{3}$

Step 5.2.2.25

Combine the numerators over the common denominator.

$\frac{- 64 + 18 \cdot 3}{3}$

Step 5.2.2.26

Simplify the numerator.

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Step 5.2.2.26.1

Multiply $18$ by $3$ .

$\frac{- 64 + 54}{3}$

Step 5.2.2.26.2

Add $- 64$ and $54$ .

$\frac{- 10}{3}$

Step 5.2.2.27

Move the negative in front of the fraction.

$- \frac{10}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{10}{3}$

Decimal Form:

$- 3.\overline{3}$

Mixed Number Form:

$- 3 \frac{1}{3}$

Step 7