Calculus Examples

$\int_{1}^{64} \frac{3 + \sqrt[3]{x}}{\sqrt{x}} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\int_{1}^{64} \frac{3 + x^{\frac{1}{3}}}{\sqrt{x}} d x$

Step 2

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

$\int_{1}^{64} \frac{3 + x^{\frac{1}{3}}}{x^{\frac{1}{2}}} d x$

Step 3

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

$\int_{1}^{64} (3 + x^{\frac{1}{3}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 4

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

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Step 4.1

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

$\int_{1}^{64} (3 + x^{\frac{1}{3}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 4.2

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

$\int_{1}^{64} (3 + x^{\frac{1}{3}}) x^{\frac{- 1}{2}} d x$

Step 4.3

Move the negative in front of the fraction.

$\int_{1}^{64} (3 + x^{\frac{1}{3}}) x^{- \frac{1}{2}} d x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{1}{3}} x^{- \frac{1}{2}} d x$

Step 5.2

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

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{1}{3} - \frac{1}{2}} d x$

Step 5.3

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2}} d x$

Step 5.4

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}} d x$

Step 5.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.1

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

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}} d x$

Step 5.5.2

Multiply $3$ by $2$ .

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{2}{6} - \frac{1}{2} \cdot \frac{3}{3}} d x$

Step 5.5.3

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

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{2}{6} - \frac{3}{2 \cdot 3}} d x$

Step 5.5.4

Multiply $2$ by $3$ .

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{2}{6} - \frac{3}{6}} d x$

Step 5.6

Combine the numerators over the common denominator.

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{2 - 3}{6}} d x$

Step 5.7

Subtract $3$ from $2$ .

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{\frac{- 1}{6}} d x$

Step 6

Move the negative in front of the fraction.

$\int_{1}^{64} 3 x^{- \frac{1}{2}} + x^{- \frac{1}{6}} d x$

Step 7

Split the single integral into multiple integrals.

$\int_{1}^{64} 3 x^{- \frac{1}{2}} d x + \int_{1}^{64} x^{- \frac{1}{6}} d x$

Step 8

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

$3 \int_{1}^{64} x^{- \frac{1}{2}} d x + \int_{1}^{64} x^{- \frac{1}{6}} d x$

Step 9

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

$3 (2 x^{\frac{1}{2}}]_{1}^{64}) + \int_{1}^{64} x^{- \frac{1}{6}} d x$

Step 10

By the Power Rule, the integral of $x^{- \frac{1}{6}}$ with respect to $x$ is $\frac{6}{5} x^{\frac{5}{6}}$ .

$3 (2 x^{\frac{1}{2}}]_{1}^{64}) + \frac{6}{5} x^{\frac{5}{6}}]_{1}^{64}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $2 x^{\frac{1}{2}}$ at $64$ and at $1$ .

$3 ((2 \cdot 64^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} x^{\frac{5}{6}}]_{1}^{64}$

Step 11.2

Evaluate $\frac{6}{5} x^{\frac{5}{6}}$ at $64$ and at $1$ .

$3 (2 \cdot 64^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3

Simplify.

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Step 11.3.1

Rewrite $64$ as $8^{2}$ .

$3 (2 \cdot {(8^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.2

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

$3 (2 \cdot 8^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.3

Cancel the common factor of $2$ .

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Step 11.3.3.1

Cancel the common factor.

$3 (2 \cdot 8^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.3.2

Rewrite the expression.

$3 (2 \cdot 8^{1} - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.4

Evaluate the exponent.

$3 (2 \cdot 8 - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.5

Multiply $2$ by $8$ .

$3 (16 - 2 \cdot 1^{\frac{1}{2}}) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.6

One to any power is one.

$3 (16 - 2 \cdot 1) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.7

Multiply $- 2$ by $1$ .

$3 (16 - 2) + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.8

Subtract $2$ from $16$ .

$3 \cdot 14 + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.9

Multiply $3$ by $14$ .

$42 + \frac{6}{5} \cdot 64^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.10

Rewrite $64$ as $2^{6}$ .

$42 + \frac{6}{5} \cdot {(2^{6})}^{\frac{5}{6}} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.11

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

$42 + \frac{6}{5} \cdot 2^{6 (\frac{5}{6})} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.12

Cancel the common factor of $6$ .

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Step 11.3.12.1

Cancel the common factor.

$42 + \frac{6}{5} \cdot 2^{6 (\frac{5}{6})} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.12.2

Rewrite the expression.

$42 + \frac{6}{5} \cdot 2^{5} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.13

Raise $2$ to the power of $5$ .

$42 + \frac{6}{5} \cdot 32 - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.14

Combine $\frac{6}{5}$ and $32$ .

$42 + \frac{6 \cdot 32}{5} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.15

Multiply $6$ by $32$ .

$42 + \frac{192}{5} - \frac{6}{5} \cdot 1^{\frac{5}{6}}$

Step 11.3.16

One to any power is one.

$42 + \frac{192}{5} - \frac{6}{5} \cdot 1$

Step 11.3.17

Multiply $- 1$ by $1$ .

$42 + \frac{192}{5} - \frac{6}{5}$

Step 11.3.18

Combine the numerators over the common denominator.

$42 + \frac{192 - 6}{5}$

Step 11.3.19

Subtract $6$ from $192$ .

$42 + \frac{186}{5}$

Step 11.3.20

To write $42$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$42 \cdot \frac{5}{5} + \frac{186}{5}$

Step 11.3.21

Combine $42$ and $\frac{5}{5}$ .

$\frac{42 \cdot 5}{5} + \frac{186}{5}$

Step 11.3.22

Combine the numerators over the common denominator.

$\frac{42 \cdot 5 + 186}{5}$

Step 11.3.23

Simplify the numerator.

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Step 11.3.23.1

Multiply $42$ by $5$ .

$\frac{210 + 186}{5}$

Step 11.3.23.2

Add $210$ and $186$ .

$\frac{396}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{396}{5}$

Decimal Form:

$79.2$

Mixed Number Form:

$79 \frac{1}{5}$

Step 13