Calculus Examples

$\int_{4}^{9} 2 x \sqrt{x} d x$

Step 1

Simplify.

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Step 1.1

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

$\int_{4}^{9} 2 x \cdot x^{\frac{1}{2}} d x$

Step 1.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.2.1

Move $x^{\frac{1}{2}}$ .

$\int_{4}^{9} 2 (x^{\frac{1}{2}} x) d x$

Step 1.2.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 1.2.2.1

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

$\int_{4}^{9} 2 (x^{\frac{1}{2}} x^{1}) d x$

Step 1.2.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Add $1$ and $2$ .

$\int_{4}^{9} 2 x^{\frac{3}{2}} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{4}^{9} x^{\frac{3}{2}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$2 (\frac{2}{5} x^{\frac{5}{2}}]_{4}^{9})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$2 (\frac{2 x^{\frac{5}{2}}}{5}]_{4}^{9})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{2 x^{\frac{5}{2}}}{5}$ at $9$ and at $4$ .

$2 ((\frac{2 \cdot 9^{\frac{5}{2}}}{5}) - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$2 (\frac{2 \cdot {(3^{2})}^{\frac{5}{2}}}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.2

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

$2 (\frac{2 \cdot 3^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.3.1

Cancel the common factor.

$2 (\frac{2 \cdot 3^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.3.2

Rewrite the expression.

$2 (\frac{2 \cdot 3^{5}}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.4

Raise $3$ to the power of $5$ .

$2 (\frac{2 \cdot 243}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.5

Multiply $2$ by $243$ .

$2 (\frac{486}{5} - \frac{2 \cdot 4^{\frac{5}{2}}}{5})$

Step 4.2.2.6

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

$2 (\frac{486}{5} - \frac{2 \cdot {(2^{2})}^{\frac{5}{2}}}{5})$

Step 4.2.2.7

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

$2 (\frac{486}{5} - \frac{2 \cdot 2^{2 (\frac{5}{2})}}{5})$

Step 4.2.2.8

Cancel the common factor of $2$ .

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Step 4.2.2.8.1

Cancel the common factor.

$2 (\frac{486}{5} - \frac{2 \cdot 2^{2 (\frac{5}{2})}}{5})$

Step 4.2.2.8.2

Rewrite the expression.

$2 (\frac{486}{5} - \frac{2 \cdot 2^{5}}{5})$

Step 4.2.2.9

Raise $2$ to the power of $5$ .

$2 (\frac{486}{5} - \frac{2 \cdot 32}{5})$

Step 4.2.2.10

Multiply $2$ by $32$ .

$2 (\frac{486}{5} - \frac{64}{5})$

Step 4.2.2.11

Combine the numerators over the common denominator.

$2 \frac{486 - 64}{5}$

Step 4.2.2.12

Subtract $64$ from $486$ .

$2 (\frac{422}{5})$

Step 4.2.2.13

Combine $2$ and $\frac{422}{5}$ .

$\frac{2 \cdot 422}{5}$

Step 4.2.2.14

Multiply $2$ by $422$ .

$\frac{844}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{844}{5}$

Decimal Form:

$168.8$

Mixed Number Form:

$168 \frac{4}{5}$

Step 6