Calculus Examples

$\int_{0}^{2} 3 \sqrt{4 u + 1} d u$

Step 1

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

$3 \int_{0}^{2} \sqrt{4 u + 1} d u$

Step 2

Let $u_{1} = 4 u + 1$ . Then $d u_{1} = 4 d u$ , so $\frac{1}{4} d u_{1} = d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 4 u + 1$ . Find $\frac{d u_{1}}{d u}$ .

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Step 2.1.1

Differentiate $4 u + 1$ .

$\frac{d}{d u} [4 u + 1]$

Step 2.1.2

By the Sum Rule, the derivative of $4 u + 1$ with respect to $u$ is $\frac{d}{d u} [4 u] + \frac{d}{d u} [1]$ .

$\frac{d}{d u} [4 u] + \frac{d}{d u} [1]$

Step 2.1.3

Evaluate $\frac{d}{d u} [4 u]$ .

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Step 2.1.3.1

Since $4$ is constant with respect to $u$ , the derivative of $4 u$ with respect to $u$ is $4 \frac{d}{d u} [u]$ .

$4 \frac{d}{d u} [u] + \frac{d}{d u} [1]$

Step 2.1.3.2

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

$4 \cdot 1 + \frac{d}{d u} [1]$

Step 2.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d u} [1]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $1$ is constant with respect to $u$ , the derivative of $1$ with respect to $u$ is $0$ .

$4 + 0$

Step 2.1.4.2

Add $4$ and $0$ .

$4$

Step 2.2

Substitute the lower limit in for $u$ in $u_{1} = 4 u + 1$ .

$u_{lower} = 4 \cdot 0 + 1$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $4$ by $0$ .

$u_{lower} = 0 + 1$

Step 2.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $u$ in $u_{1} = 4 u + 1$ .

$u_{upper} = 4 \cdot 2 + 1$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $4$ by $2$ .

$u_{upper} = 8 + 1$

Step 2.5.2

Add $8$ and $1$ .

$u_{upper} = 9$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 2.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$3 \int_{1}^{9} \sqrt{u_{1}} \frac{1}{4} d u_{1}$

Step 3

Combine $\sqrt{u_{1}}$ and $\frac{1}{4}$ .

$3 \int_{1}^{9} \frac{\sqrt{u_{1}}}{4} d u_{1}$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} \int_{1}^{9} \sqrt{u_{1}} d u_{1})$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int_{1}^{9} \sqrt{u_{1}} d u_{1}$

Step 5.2

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

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

Step 6

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

$\frac{3}{4} \frac{2}{3} {u_{1}}^{\frac{3}{2}}]_{1}^{9}$

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor of $2$ .

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Step 7.2.3.1

Cancel the common factor.

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

Step 7.2.3.2

Rewrite the expression.

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

Step 7.2.4

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

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

Step 7.2.5

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

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

Step 7.2.6

Multiply $2$ by $27$ .

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

Step 7.2.7

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

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Step 7.2.7.1

Factor $3$ out of $54$ .

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

Step 7.2.7.2

Cancel the common factors.

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Step 7.2.7.2.1

Factor $3$ out of $3$ .

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

Step 7.2.7.2.2

Cancel the common factor.

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

Step 7.2.7.2.3

Rewrite the expression.

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

Step 7.2.7.2.4

Divide $18$ by $1$ .

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

Step 7.2.8

One to any power is one.

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

Step 7.2.9

Multiply $- 1$ by $1$ .

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

Step 7.2.10

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

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

Step 7.2.11

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

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

Step 7.2.12

Combine the numerators over the common denominator.

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

Step 7.2.13

Simplify the numerator.

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Step 7.2.13.1

Multiply $18$ by $3$ .

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

Step 7.2.13.2

Subtract $2$ from $54$ .

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

Step 7.2.14

Multiply $\frac{3}{4}$ by $\frac{52}{3}$ .

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

Step 7.2.15

Multiply $3$ by $52$ .

$\frac{156}{4 \cdot 3}$

Step 7.2.16

Multiply $4$ by $3$ .

$\frac{156}{12}$

Step 7.2.17

Cancel the common factor of $156$ and $12$ .

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Step 7.2.17.1

Factor $12$ out of $156$ .

$\frac{12 \cdot 13}{12}$

Step 7.2.17.2

Cancel the common factors.

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Step 7.2.17.2.1

Factor $12$ out of $12$ .

$\frac{12 \cdot 13}{12 (1)}$

Step 7.2.17.2.2

Cancel the common factor.

$\frac{12 \cdot 13}{12 \cdot 1}$

Step 7.2.17.2.3

Rewrite the expression.

$\frac{13}{1}$

Step 7.2.17.2.4

Divide $13$ by $1$ .

$13$

Step 8