Calculus Examples

$\int_{4}^{9} 3 \sqrt{x} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{4}^{9} \sqrt{x} d x$

Step 2

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

$3 \int_{4}^{9} x^{\frac{1}{2}} d x$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

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 $4$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

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.

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

Step 4.2.2.3.2

Rewrite the expression.

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

Step 4.2.2.4

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

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

Step 4.2.2.5

Multiply $2$ by $27$ .

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

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 $54$ .

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

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 $3$ .

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

Step 4.2.2.6.2.2

Cancel the common factor.

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

Step 4.2.2.6.2.3

Rewrite the expression.

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

Step 4.2.2.6.2.4

Divide $18$ by $1$ .

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

Step 4.2.2.7

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

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

Step 4.2.2.8

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

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

Step 4.2.2.9

Cancel the common factor of $2$ .

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Step 4.2.2.9.1

Cancel the common factor.

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

Step 4.2.2.9.2

Rewrite the expression.

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

Step 4.2.2.10

Raise $2$ to the power of $3$ .

$3 (18 - \frac{2 \cdot 8}{3})$

Step 4.2.2.11

Multiply $2$ by $8$ .

$3 (18 - \frac{16}{3})$

Step 4.2.2.12

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

$3 (18 \cdot \frac{3}{3} - \frac{16}{3})$

Step 4.2.2.13

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

$3 (\frac{18 \cdot 3}{3} - \frac{16}{3})$

Step 4.2.2.14

Combine the numerators over the common denominator.

$3 \frac{18 \cdot 3 - 16}{3}$

Step 4.2.2.15

Simplify the numerator.

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Step 4.2.2.15.1

Multiply $18$ by $3$ .

$3 \frac{54 - 16}{3}$

Step 4.2.2.15.2

Subtract $16$ from $54$ .

$3 (\frac{38}{3})$

Step 4.2.2.16

Combine $3$ and $\frac{38}{3}$ .

$\frac{3 \cdot 38}{3}$

Step 4.2.2.17

Multiply $3$ by $38$ .

$\frac{114}{3}$

Step 4.2.2.18

Cancel the common factor of $114$ and $3$ .

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Step 4.2.2.18.1

Factor $3$ out of $114$ .

$\frac{3 \cdot 38}{3}$

Step 4.2.2.18.2

Cancel the common factors.

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Step 4.2.2.18.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 38}{3 (1)}$

Step 4.2.2.18.2.2

Cancel the common factor.

$\frac{3 \cdot 38}{3 \cdot 1}$

Step 4.2.2.18.2.3

Rewrite the expression.

$\frac{38}{1}$

Step 4.2.2.18.2.4

Divide $38$ by $1$ .

$38$

Step 5