Calculus Examples

$\frac{d z}{\sqrt[4]{z}}$

Step 1

Since $\frac{z}{\sqrt[4]{z}}$ is constant with respect to $d$ , move $\frac{z}{\sqrt[4]{z}}$ out of the integral.

$\frac{z}{\sqrt[4]{z}} \int d d \cdot d$

Step 2

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

$\frac{z}{\sqrt[4]{z}} (\frac{1}{2} d^{2} + C)$

Step 3

Simplify the answer.

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Step 3.1

Rewrite $\frac{z}{\sqrt[4]{z}} (\frac{1}{2} d^{2} + C)$ as $\frac{z {\sqrt[4]{z}}^{3}}{z} (\frac{1}{2} d^{2}) + C$ .

$\frac{z {\sqrt[4]{z}}^{3}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite ${\sqrt[4]{z}}^{3}$ as $\sqrt[4]{z^{3}}$ .

$\frac{z \sqrt[4]{z^{3}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{z^{3}}$ as $z^{\frac{3}{4}}$ .

$\frac{z \cdot z^{\frac{3}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.3

Multiply $z$ by $z^{\frac{3}{4}}$ by adding the exponents.

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Step 3.2.3.1

Multiply $z$ by $z^{\frac{3}{4}}$ .

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Step 3.2.3.1.1

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

$\frac{z^{1} z^{\frac{3}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.3.1.2

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

$\frac{z^{1 + \frac{3}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.3.2

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

$\frac{z^{\frac{4}{4} + \frac{3}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.3.3

Combine the numerators over the common denominator.

$\frac{z^{\frac{4 + 3}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.3.4

Add $4$ and $3$ .

$\frac{z^{\frac{7}{4}}}{z} (\frac{1}{2} d^{2}) + C$

Step 3.2.4

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

$z^{\frac{7}{4}} z^{- 1} (\frac{1}{2} d^{2}) + C$

Step 3.2.5

Multiply $z^{\frac{7}{4}}$ by $z^{- 1}$ by adding the exponents.

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Step 3.2.5.1

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

$z^{\frac{7}{4} - 1} (\frac{1}{2} d^{2}) + C$

Step 3.2.5.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$z^{\frac{7}{4} - 1 \cdot \frac{4}{4}} (\frac{1}{2} d^{2}) + C$

Step 3.2.5.3

Combine $- 1$ and $\frac{4}{4}$ .

$z^{\frac{7}{4} + \frac{- 1 \cdot 4}{4}} (\frac{1}{2} d^{2}) + C$

Step 3.2.5.4

Combine the numerators over the common denominator.

$z^{\frac{7 - 1 \cdot 4}{4}} (\frac{1}{2} d^{2}) + C$

Step 3.2.5.5

Simplify the numerator.

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Step 3.2.5.5.1

Multiply $- 1$ by $4$ .

$z^{\frac{7 - 4}{4}} (\frac{1}{2} d^{2}) + C$

Step 3.2.5.5.2

Subtract $4$ from $7$ .

$z^{\frac{3}{4}} (\frac{1}{2} d^{2}) + C$

Step 3.2.6

Combine $\frac{1}{2}$ and $d^{2}$ .

$z^{\frac{3}{4}} \frac{d^{2}}{2} + C$

Step 3.2.7

Combine $z^{\frac{3}{4}}$ and $\frac{d^{2}}{2}$ .

$\frac{z^{\frac{3}{4}} d^{2}}{2} + C$

$\frac{1}{2} z^{\frac{3}{4}} d^{2} + C$