Calculus Examples

$\int_{1}^{9} 4 \sqrt{x} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{9} \sqrt{x} d x$

Step 2

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

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

Step 3

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

$4 (\frac{2}{3} x^{\frac{3}{2}}]_{1}^{9})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$4 (\frac{2 x^{\frac{3}{2}}}{3}]_{1}^{9})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

$4 ((\frac{2 \cdot 9^{\frac{3}{2}}}{3}) - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$4 (\frac{2 \cdot {(3^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.2

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

$4 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

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.

$4 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.3.2

Rewrite the expression.

$4 (\frac{2 \cdot 3^{3}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.4

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

$4 (\frac{2 \cdot 27}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.5

Multiply $2$ by $27$ .

$4 (\frac{54}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.6

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

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Step 4.2.2.6.1

Factor $3$ out of $54$ .

$4 (\frac{3 \cdot 18}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.6.2

Cancel the common factors.

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Step 4.2.2.6.2.1

Factor $3$ out of $3$ .

$4 (\frac{3 \cdot 18}{3 (1)} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.6.2.2

Cancel the common factor.

$4 (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.6.2.3

Rewrite the expression.

$4 (\frac{18}{1} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.6.2.4

Divide $18$ by $1$ .

$4 (18 - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.7

One to any power is one.

$4 (18 - \frac{2 \cdot 1}{3})$

Step 4.2.2.8

Multiply $2$ by $1$ .

$4 (18 - \frac{2}{3})$

Step 4.2.2.9

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

$4 (18 \cdot \frac{3}{3} - \frac{2}{3})$

Step 4.2.2.10

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

$4 (\frac{18 \cdot 3}{3} - \frac{2}{3})$

Step 4.2.2.11

Combine the numerators over the common denominator.

$4 \frac{18 \cdot 3 - 2}{3}$

Step 4.2.2.12

Simplify the numerator.

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Step 4.2.2.12.1

Multiply $18$ by $3$ .

$4 \frac{54 - 2}{3}$

Step 4.2.2.12.2

Subtract $2$ from $54$ .

$4 (\frac{52}{3})$

Step 4.2.2.13

Combine $4$ and $\frac{52}{3}$ .

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

Step 4.2.2.14

Multiply $4$ by $52$ .

$\frac{208}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{208}{3}$

Decimal Form:

$69.\overline{3}$

Mixed Number Form:

$69 \frac{1}{3}$

Step 6