Calculus Examples

$\int_{1}^{6} x \sqrt{5 x^{2} + 4} d x$

Step 1

Let $u = 5 x^{2} + 4$ . Then $d u = 10 x d x$ , so $\frac{1}{10} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 5 x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $5 x^{2} + 4$ .

$\frac{d}{d x} [5 x^{2} + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $5 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$5 (2 x) + \frac{d}{d x} [4]$

Step 1.1.3.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [4]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$10 x + 0$

Step 1.1.4.2

Add $10 x$ and $0$ .

$10 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 5 x^{2} + 4$ .

$u_{lower} = 5 \cdot 1^{2} + 4$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

One to any power is one.

$u_{lower} = 5 \cdot 1 + 4$

Step 1.3.1.2

Multiply $5$ by $1$ .

$u_{lower} = 5 + 4$

Step 1.3.2

Add $5$ and $4$ .

$u_{lower} = 9$

Step 1.4

Substitute the upper limit in for $x$ in $u = 5 x^{2} + 4$ .

$u_{upper} = 5 \cdot 6^{2} + 4$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $6$ to the power of $2$ .

$u_{upper} = 5 \cdot 36 + 4$

Step 1.5.1.2

Multiply $5$ by $36$ .

$u_{upper} = 180 + 4$

Step 1.5.2

Add $180$ and $4$ .

$u_{upper} = 184$

Step 1.6

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

$u_{lower} = 9$

$u_{upper} = 184$

Step 1.7

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

$\int_{9}^{184} \sqrt{u} \frac{1}{10} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{10}$ .

$\int_{9}^{184} \frac{\sqrt{u}}{10} d u$

Step 3

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$\frac{1}{10} \int_{9}^{184} \sqrt{u} d u$

Step 4

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

$\frac{1}{10} \int_{9}^{184} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{10} \frac{2}{3} u^{\frac{3}{2}}]_{9}^{184}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $184$ and at $9$ .

$\frac{1}{10} ((\frac{2}{3} \cdot 184^{\frac{3}{2}}) - \frac{2}{3} \cdot 9^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{2}{3}$ and $184^{\frac{3}{2}}$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 9^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}})$

Step 6.2.3

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

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})})$

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})})$

Step 6.2.4.2

Rewrite the expression.

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{3})$

Step 6.2.5

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

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 27)$

Step 6.2.6

Multiply $27$ by $- 1$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - 27 (\frac{2}{3}))$

Step 6.2.7

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

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{- 27 \cdot 2}{3})$

Step 6.2.8

Multiply $- 27$ by $2$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{- 54}{3})$

Step 6.2.9

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

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Step 6.2.9.1

Factor $3$ out of $- 54$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3})$

Step 6.2.9.2

Cancel the common factors.

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Step 6.2.9.2.1

Factor $3$ out of $3$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3 (1)})$

Step 6.2.9.2.2

Cancel the common factor.

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3 \cdot 1})$

Step 6.2.9.2.3

Rewrite the expression.

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{- 18}{1})$

Step 6.2.9.2.4

Divide $- 18$ by $1$ .

$\frac{1}{10} (\frac{2 \cdot 184^{\frac{3}{2}}}{3} - 18)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{10} \cdot \frac{2 \cdot 184^{\frac{3}{2}}}{3} + \frac{1}{10} \cdot - 18$

Step 7.2

Combine.

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{1}{10} \cdot - 18$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Factor $2$ out of $10$ .

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{1}{2 (5)} \cdot - 18$

Step 7.3.2

Factor $2$ out of $- 18$ .

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{1}{2 \cdot 5} \cdot (2 \cdot - 9)$

Step 7.3.3

Cancel the common factor.

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{1}{2 \cdot 5} \cdot (2 \cdot - 9)$

Step 7.3.4

Rewrite the expression.

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{1}{5} \cdot - 9$

Step 7.4

Combine $\frac{1}{5}$ and $- 9$ .

$\frac{1 (2 \cdot 184^{\frac{3}{2}})}{10 \cdot 3} + \frac{- 9}{5}$

Step 7.5

Simplify each term.

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Step 7.5.1

Factor $2$ out of $1 (2 \cdot 184^{\frac{3}{2}})$ .

$\frac{2 (1 \cdot (184^{\frac{3}{2}}))}{10 \cdot 3} + \frac{- 9}{5}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $2$ out of $10 \cdot 3$ .

$\frac{2 (1 \cdot (184^{\frac{3}{2}}))}{2 (5 \cdot 3)} + \frac{- 9}{5}$

Step 7.5.2.2

Cancel the common factor.

$\frac{2 (1 \cdot (184^{\frac{3}{2}}))}{2 (5 \cdot 3)} + \frac{- 9}{5}$

Step 7.5.2.3

Rewrite the expression.

$\frac{1 \cdot (184^{\frac{3}{2}})}{5 \cdot 3} + \frac{- 9}{5}$

Step 7.5.3

Multiply $184^{\frac{3}{2}}$ by $1$ .

$\frac{184^{\frac{3}{2}}}{5 \cdot 3} + \frac{- 9}{5}$

Step 7.5.4

Multiply $5$ by $3$ .

$\frac{184^{\frac{3}{2}}}{15} + \frac{- 9}{5}$

Step 7.5.5

Move the negative in front of the fraction.

$\frac{184^{\frac{3}{2}}}{15} - \frac{9}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{184^{\frac{3}{2}}}{15} - \frac{9}{5}$

Decimal Form:

$164.59316225 \dots$

Step 9