Calculus Examples

$\int_{0}^{27} \sqrt{3 x} d x$

Step 1

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 1.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \cdot 27$

Step 1.5

Multiply $3$ by $27$ .

$u_{upper} = 81$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 81$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{81} \sqrt{u} \frac{1}{3} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$\int_{0}^{81} \frac{\sqrt{u}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{0}^{81} \sqrt{u} d u$

Step 4

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

$\frac{1}{3} \int_{0}^{81} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{3} \frac{2}{3} u^{\frac{3}{2}}]_{0}^{81}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $81$ and at $0$ .

$\frac{1}{3} ((\frac{2}{3} \cdot 81^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $81$ as $9^{2}$ .

$\frac{1}{3} (\frac{2}{3} \cdot {(9^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{3} (\frac{2}{3} \cdot 9^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{3} (\frac{2}{3} \cdot 9^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{3} (\frac{2}{3} \cdot 9^{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.4

Raise $9$ to the power of $3$ .

$\frac{1}{3} (\frac{2}{3} \cdot 729 - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.5

Combine $\frac{2}{3}$ and $729$ .

$\frac{1}{3} (\frac{2 \cdot 729}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.6

Multiply $2$ by $729$ .

$\frac{1}{3} (\frac{1458}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7

Cancel the common factor of $1458$ and $3$ .

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Step 6.2.7.1

Factor $3$ out of $1458$ .

$\frac{1}{3} (\frac{3 \cdot 486}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7.2

Cancel the common factors.

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Step 6.2.7.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} (\frac{3 \cdot 486}{3 (1)} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7.2.2

Cancel the common factor.

$\frac{1}{3} (\frac{3 \cdot 486}{3 \cdot 1} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7.2.3

Rewrite the expression.

$\frac{1}{3} (\frac{486}{1} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7.2.4

Divide $486$ by $1$ .

$\frac{1}{3} (486 - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.8

Rewrite $0$ as $0^{2}$ .

$\frac{1}{3} (486 - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 6.2.9

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

$\frac{1}{3} (486 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.10

Cancel the common factor of $2$ .

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Step 6.2.10.1

Cancel the common factor.

$\frac{1}{3} (486 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.10.2

Rewrite the expression.

$\frac{1}{3} (486 - \frac{2}{3} \cdot 0^{3})$

Step 6.2.11

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} (486 - \frac{2}{3} \cdot 0)$

Step 6.2.12

Multiply $0$ by $- 1$ .

$\frac{1}{3} (486 + 0 (\frac{2}{3}))$

Step 6.2.13

Multiply $0$ by $\frac{2}{3}$ .

$\frac{1}{3} (486 + 0)$

Step 6.2.14

Add $486$ and $0$ .

$\frac{1}{3} \cdot 486$

Step 6.2.15

Combine $\frac{1}{3}$ and $486$ .

$\frac{486}{3}$

Step 6.2.16

Cancel the common factor of $486$ and $3$ .

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Step 6.2.16.1

Factor $3$ out of $486$ .

$\frac{3 \cdot 162}{3}$

Step 6.2.16.2

Cancel the common factors.

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Step 6.2.16.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 162}{3 (1)}$

Step 6.2.16.2.2

Cancel the common factor.

$\frac{3 \cdot 162}{3 \cdot 1}$

Step 6.2.16.2.3

Rewrite the expression.

$\frac{162}{1}$

Step 6.2.16.2.4

Divide $162$ by $1$ .

$162$

Step 7