Calculus Examples

$\int_{1}^{16} 5 + x^{\frac{1}{4}} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{16} 5 d x + \int_{1}^{16} x^{\frac{1}{4}} d x$

Step 2

Apply the constant rule.

$5 x]_{1}^{16} + \int_{1}^{16} x^{\frac{1}{4}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$5 x]_{1}^{16} + \frac{4}{5} x^{\frac{5}{4}}]_{1}^{16}$

Step 4

Simplify the answer.

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Step 4.1

Combine $5 x]_{1}^{16}$ and $\frac{4}{5} x^{\frac{5}{4}}]_{1}^{16}$ .

$5 x + \frac{4}{5} x^{\frac{5}{4}}]_{1}^{16}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $5 x + \frac{4}{5} x^{\frac{5}{4}}$ at $16$ and at $1$ .

$(5 \cdot 16 + \frac{4}{5} \cdot 16^{\frac{5}{4}}) - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $5$ by $16$ .

$80 + \frac{4}{5} \cdot 16^{\frac{5}{4}} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.2

Rewrite $16$ as $2^{4}$ .

$80 + \frac{4}{5} \cdot {(2^{4})}^{\frac{5}{4}} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.3

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

$80 + \frac{4}{5} \cdot 2^{4 (\frac{5}{4})} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.4

Cancel the common factor of $4$ .

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Step 4.2.2.4.1

Cancel the common factor.

$80 + \frac{4}{5} \cdot 2^{4 (\frac{5}{4})} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.4.2

Rewrite the expression.

$80 + \frac{4}{5} \cdot 2^{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.5

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

$80 + \frac{4}{5} \cdot 32 - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.6

Combine $\frac{4}{5}$ and $32$ .

$80 + \frac{4 \cdot 32}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.7

Multiply $4$ by $32$ .

$80 + \frac{128}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.8

To write $80$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$80 \cdot \frac{5}{5} + \frac{128}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.9

Combine $80$ and $\frac{5}{5}$ .

$\frac{80 \cdot 5}{5} + \frac{128}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.10

Combine the numerators over the common denominator.

$\frac{80 \cdot 5 + 128}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.11

Simplify the numerator.

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Step 4.2.2.11.1

Multiply $80$ by $5$ .

$\frac{400 + 128}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.11.2

Add $400$ and $128$ .

$\frac{528}{5} - (5 \cdot 1 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.12

Multiply $5$ by $1$ .

$\frac{528}{5} - (5 + \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 4.2.2.13

One to any power is one.

$\frac{528}{5} - (5 + \frac{4}{5} \cdot 1)$

Step 4.2.2.14

Multiply $\frac{4}{5}$ by $1$ .

$\frac{528}{5} - (5 + \frac{4}{5})$

Step 4.2.2.15

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

$\frac{528}{5} - (5 \cdot \frac{5}{5} + \frac{4}{5})$

Step 4.2.2.16

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

$\frac{528}{5} - (\frac{5 \cdot 5}{5} + \frac{4}{5})$

Step 4.2.2.17

Combine the numerators over the common denominator.

$\frac{528}{5} - \frac{5 \cdot 5 + 4}{5}$

Step 4.2.2.18

Simplify the numerator.

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Step 4.2.2.18.1

Multiply $5$ by $5$ .

$\frac{528}{5} - \frac{25 + 4}{5}$

Step 4.2.2.18.2

Add $25$ and $4$ .

$\frac{528}{5} - \frac{29}{5}$

Step 4.2.2.19

Combine the numerators over the common denominator.

$\frac{528 - 29}{5}$

Step 4.2.2.20

Subtract $29$ from $528$ .

$\frac{499}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{499}{5}$

Decimal Form:

$99.8$

Mixed Number Form:

$99 \frac{4}{5}$

Step 6