Calculus Examples

$\int_{1}^{5} \frac{x}{\sqrt{2 x - 1}} d x$

Step 1

Let $u = 2 x - 1$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

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

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Step 1.1.3.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x - 1$ .

$u_{lower} = 2 \cdot 1 - 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $1$ .

$u_{lower} = 2 - 1$

Step 1.3.2

Subtract $1$ from $2$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x - 1$ .

$u_{upper} = 2 \cdot 5 - 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $5$ .

$u_{upper} = 10 - 1$

Step 1.5.2

Subtract $1$ from $10$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 1.7

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

$\int_{1}^{9} \frac{\frac{u}{2} + \frac{1}{2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{\frac{u}{2} + \frac{1}{2}}{\sqrt{u}}$ by $\frac{2}{2}$ .

$\int_{1}^{9} \frac{2}{2} \cdot \frac{\frac{u}{2} + \frac{1}{2}}{\sqrt{u}} \frac{1}{2} d u$

Step 2.2

Combine.

$\int_{1}^{9} \frac{2 (\frac{u}{2} + \frac{1}{2})}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.3

Apply the distributive property.

$\int_{1}^{9} \frac{2 \frac{u}{2} + 2 (\frac{1}{2})}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\int_{1}^{9} \frac{2 \frac{u}{2} + 2 (\frac{1}{2})}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.4.2

Rewrite the expression.

$\int_{1}^{9} \frac{u + 2 (\frac{1}{2})}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

$\int_{1}^{9} \frac{u + 2 (\frac{1}{2})}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.5.2

Rewrite the expression.

$\int_{1}^{9} \frac{u + 1}{2 \sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.6

Multiply $\frac{u + 1}{2 \sqrt{u}}$ by $\frac{1}{2}$ .

$\int_{1}^{9} \frac{u + 1}{2 \sqrt{u} \cdot 2} d u$

Step 2.7

Multiply $2$ by $2$ .

$\int_{1}^{9} \frac{u + 1}{4 \sqrt{u}} d u$

Step 3

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

$\frac{1}{4} \int_{1}^{9} \frac{u + 1}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{4} \int_{1}^{9} (u + 1) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 4.3.1

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

$\frac{1}{4} \int_{1}^{9} (u + 1) u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{1}{4} \int_{1}^{9} (u + 1) u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{4} \int_{1}^{9} (u + 1) u^{- \frac{1}{2}} d u$

Step 5

Expand $(u + 1) u^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\frac{1}{4} \int_{1}^{9} u \cdot u^{- \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.2

Raise $u$ to the power of $1$ .

$\frac{1}{4} \int_{1}^{9} u^{1} u^{- \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{4} \int_{1}^{9} u^{1 - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.4

Write $1$ as a fraction with a common denominator.

$\frac{1}{4} \int_{1}^{9} u^{\frac{2}{2} - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.5

Combine the numerators over the common denominator.

$\frac{1}{4} \int_{1}^{9} u^{\frac{2 - 1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.6

Subtract $1$ from $2$ .

$\frac{1}{4} \int_{1}^{9} u^{\frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 5.7

Multiply $u^{- \frac{1}{2}}$ by $1$ .

$\frac{1}{4} \int_{1}^{9} u^{\frac{1}{2}} + u^{- \frac{1}{2}} d u$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{4} (\int_{1}^{9} u^{\frac{1}{2}} d u + \int_{1}^{9} u^{- \frac{1}{2}} d u)$

Step 7

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

$\frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9} + \int_{1}^{9} u^{- \frac{1}{2}} d u)$

Step 8

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

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

Step 9

Combine $\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}$ and $2 u^{\frac{1}{2}}]_{1}^{9}$ .

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

Cancel the common factor of $2$ .

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Step 11.3.1

Cancel the common factor.

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

Step 11.3.2

Rewrite the expression.

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

Step 11.4

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

$\frac{1}{4} (\frac{2}{3} \cdot 27 + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12

Simplify.

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Step 12.1

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

$\frac{1}{4} (\frac{2 \cdot 27}{3} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.2

Multiply $2$ by $27$ .

$\frac{1}{4} (\frac{54}{3} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.3

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

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Step 12.3.1

Factor $3$ out of $54$ .

$\frac{1}{4} (\frac{3 \cdot 18}{3} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} (\frac{3 \cdot 18}{3 (1)} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.3.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{3 \cdot 18}{3 \cdot 1} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.3.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{18}{1} + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 12.3.2.4

Divide $18$ by $1$ .

$\frac{1}{4} (18 + 2 \cdot 9^{\frac{1}{2}} - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

Cancel the common factor of $2$ .

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Step 13.3.1

Cancel the common factor.

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

Step 13.3.2

Rewrite the expression.

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

Step 13.4

Evaluate the exponent.

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

Step 14

Simplify the expression.

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Step 14.1

Multiply $2$ by $3$ .

$\frac{1}{4} (18 + 6 - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 14.2

Add $18$ and $6$ .

$\frac{1}{4} (24 - (\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}}))$

Step 14.3

One to any power is one.

$\frac{1}{4} (24 - (\frac{2}{3} \cdot 1 + 2 \cdot 1^{\frac{1}{2}}))$

Step 14.4

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

$\frac{1}{4} (24 - (\frac{2}{3} + 2 \cdot 1^{\frac{1}{2}}))$

Step 14.5

One to any power is one.

$\frac{1}{4} (24 - (\frac{2}{3} + 2 \cdot 1))$

Step 14.6

Multiply $2$ by $1$ .

$\frac{1}{4} (24 - (\frac{2}{3} + 2))$

Step 15

Simplify.

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Step 15.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{4} (24 - (\frac{2}{3} + 2 \cdot \frac{3}{3}))$

Step 15.2

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

$\frac{1}{4} (24 - (\frac{2}{3} + \frac{2 \cdot 3}{3}))$

Step 15.3

Combine the numerators over the common denominator.

$\frac{1}{4} (24 - \frac{2 + 2 \cdot 3}{3})$

Step 15.4

Simplify the numerator.

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Step 15.4.1

Multiply $2$ by $3$ .

$\frac{1}{4} (24 - \frac{2 + 6}{3})$

Step 15.4.2

Add $2$ and $6$ .

$\frac{1}{4} (24 - \frac{8}{3})$

Step 16

Simplify.

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Step 16.1

To write $24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{4} (24 \cdot \frac{3}{3} - \frac{8}{3})$

Step 16.2

Combine $24$ and $\frac{3}{3}$ .

$\frac{1}{4} (\frac{24 \cdot 3}{3} - \frac{8}{3})$

Step 16.3

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{24 \cdot 3 - 8}{3}$

Step 16.4

Simplify the numerator.

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Step 16.4.1

Multiply $24$ by $3$ .

$\frac{1}{4} \cdot \frac{72 - 8}{3}$

Step 16.4.2

Subtract $8$ from $72$ .

$\frac{1}{4} \cdot \frac{64}{3}$

Step 17

Simplify.

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Step 17.1

Multiply $\frac{1}{4}$ by $\frac{64}{3}$ .

$\frac{64}{4 \cdot 3}$

Step 17.2

Multiply $4$ by $3$ .

$\frac{64}{12}$

Step 17.3

Cancel the common factor of $64$ and $12$ .

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Step 17.3.1

Factor $4$ out of $64$ .

$\frac{4 (16)}{12}$

Step 17.3.2

Cancel the common factors.

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Step 17.3.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 16}{4 \cdot 3}$

Step 17.3.2.2

Cancel the common factor.

$\frac{4 \cdot 16}{4 \cdot 3}$

Step 17.3.2.3

Rewrite the expression.

$\frac{16}{3}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{3}$

Decimal Form:

$5.\overline{3}$

Mixed Number Form:

$5 \frac{1}{3}$