Calculus Examples

$\int_{1}^{4} \frac{(\sqrt{2 + \sqrt{x}})}{\sqrt{x}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

$\int_{1}^{4} \frac{{(2 + x^{\frac{1}{2}})}^{\frac{1}{2}}}{\sqrt{x}} d x$

Step 1.3

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

$\int_{1}^{4} \frac{{(2 + x^{\frac{1}{2}})}^{\frac{1}{2}}}{x^{\frac{1}{2}}} d x$

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $2 + x^{\frac{1}{2}}$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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Step 2.1.3.1

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

$0 + \frac{1}{2} x^{\frac{1}{2} - 1}$

Step 2.1.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 + \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.1.3.3

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

$0 + \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.1.3.4

Combine the numerators over the common denominator.

$0 + \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.1.3.5

Simplify the numerator.

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Step 2.1.3.5.1

Multiply $- 1$ by $2$ .

$0 + \frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 2.1.3.5.2

Subtract $2$ from $1$ .

$0 + \frac{1}{2} x^{\frac{- 1}{2}}$

Step 2.1.3.6

Move the negative in front of the fraction.

$0 + \frac{1}{2} x^{- \frac{1}{2}}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 2.1.4.2

Combine terms.

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Step 2.1.4.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$0 + \frac{1}{2 x^{\frac{1}{2}}}$

Step 2.1.4.2.2

Add $0$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 2.2

Substitute the lower limit in for $x$ in $u = 2 + x^{\frac{1}{2}}$ .

$u_{lower} = 2 + 1^{\frac{1}{2}}$

Step 2.3

Simplify.

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Step 2.3.1

One to any power is one.

$u_{lower} = 2 + 1$

Step 2.3.2

Add $2$ and $1$ .

$u_{lower} = 3$

Step 2.4

Substitute the upper limit in for $x$ in $u = 2 + x^{\frac{1}{2}}$ .

$u_{upper} = 2 + 4^{\frac{1}{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$u_{upper} = 2 + {(2^{2})}^{\frac{1}{2}}$

Step 2.5.1.2

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

$u_{upper} = 2 + 2^{2 (\frac{1}{2})}$

Step 2.5.1.3

Cancel the common factor of $2$ .

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Step 2.5.1.3.1

Cancel the common factor.

$u_{upper} = 2 + 2^{2 (\frac{1}{2})}$

Step 2.5.1.3.2

Rewrite the expression.

$u_{upper} = 2 + 2^{1}$

Step 2.5.1.4

Evaluate the exponent.

$u_{upper} = 2 + 2$

Step 2.5.2

Add $2$ and $2$ .

$u_{upper} = 4$

Step 2.6

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

$u_{lower} = 3$

$u_{upper} = 4$

Step 2.7

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

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

Step 3

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2 u^{\frac{3}{2}}}{3}$ at $4$ and at $3$ .

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

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

Step 6.2.3.2

Rewrite the expression.

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

Step 6.2.4

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

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

Step 6.2.5

Multiply $2$ by $8$ .

$2 (\frac{16}{3} - \frac{2 \cdot 3^{\frac{3}{2}}}{3})$

Step 6.2.6

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 (\frac{16}{3} - (2 \cdot 3^{\frac{3}{2}} \cdot 3^{- 1}))$

Step 6.2.7

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

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Step 6.2.7.1

Move $3^{- 1}$ .

$2 (\frac{16}{3} - (2 \cdot (3^{- 1} \cdot 3^{\frac{3}{2}})))$

Step 6.2.7.2

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

$2 (\frac{16}{3} - (2 \cdot 3^{- 1 + \frac{3}{2}}))$

Step 6.2.7.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (\frac{16}{3} - (2 \cdot 3^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}))$

Step 6.2.7.4

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

$2 (\frac{16}{3} - (2 \cdot 3^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}))$

Step 6.2.7.5

Combine the numerators over the common denominator.

$2 (\frac{16}{3} - (2 \cdot 3^{\frac{- 1 \cdot 2 + 3}{2}}))$

Step 6.2.7.6

Simplify the numerator.

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Step 6.2.7.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{16}{3} - (2 \cdot 3^{\frac{- 2 + 3}{2}}))$

Step 6.2.7.6.2

Add $- 2$ and $3$ .

$2 (\frac{16}{3} - (2 \cdot 3^{\frac{1}{2}}))$

Step 6.2.8

Multiply $2$ by $- 1$ .

$2 (\frac{16}{3} - 2 \cdot 3^{\frac{1}{2}})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (\frac{16}{3}) + 2 (- 2 \cdot 3^{\frac{1}{2}})$

Step 7.2

Multiply $2 (\frac{16}{3})$ .

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Step 7.2.1

Combine $2$ and $\frac{16}{3}$ .

$\frac{2 \cdot 16}{3} + 2 (- 2 \cdot 3^{\frac{1}{2}})$

Step 7.2.2

Multiply $2$ by $16$ .

$\frac{32}{3} + 2 (- 2 \cdot 3^{\frac{1}{2}})$

Step 7.3

Multiply $- 2$ by $2$ .

$\frac{32}{3} - 4 \cdot 3^{\frac{1}{2}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3} - 4 \cdot 3^{\frac{1}{2}}$

Decimal Form:

$3.73846343 \dots$

Step 9