Calculus Examples

$\int \frac{1 - 2 \sqrt[3]{z}}{\sqrt[3]{z}} d z$

Step 1

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

$\int \frac{1 - 2 z^{\frac{1}{3}}}{\sqrt[3]{z}} d z$

Step 2

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

$\int \frac{1 - 2 z^{\frac{1}{3}}}{z^{\frac{1}{3}}} d z$

Step 3

Move $z^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int (1 - 2 z^{\frac{1}{3}}) {(z^{\frac{1}{3}})}^{- 1} d z$

Step 4

Multiply the exponents in ${(z^{\frac{1}{3}})}^{- 1}$ .

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Step 4.1

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

$\int (1 - 2 z^{\frac{1}{3}}) z^{\frac{1}{3} \cdot - 1} d z$

Step 4.2

Combine $\frac{1}{3}$ and $- 1$ .

$\int (1 - 2 z^{\frac{1}{3}}) z^{\frac{- 1}{3}} d z$

Step 4.3

Move the negative in front of the fraction.

$\int (1 - 2 z^{\frac{1}{3}}) z^{- \frac{1}{3}} d z$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\int 1 z^{- \frac{1}{3}} - 2 z^{\frac{1}{3}} z^{- \frac{1}{3}} d z$

Step 5.2

Multiply $z^{- \frac{1}{3}}$ by $1$ .

$\int z^{- \frac{1}{3}} - 2 z^{\frac{1}{3}} z^{- \frac{1}{3}} d z$

Step 5.3

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

$\int z^{- \frac{1}{3}} - 2 z^{\frac{1}{3} - \frac{1}{3}} d z$

Step 5.4

Combine the numerators over the common denominator.

$\int z^{- \frac{1}{3}} - 2 z^{\frac{1 - 1}{3}} d z$

Step 5.5

Subtract $1$ from $1$ .

$\int z^{- \frac{1}{3}} - 2 z^{\frac{0}{3}} d z$

Step 5.6

Cancel the common factor of $0$ and $3$ .

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Step 5.6.1

Factor $3$ out of $0$ .

$\int z^{- \frac{1}{3}} - 2 z^{\frac{3 (0)}{3}} d z$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $3$ out of $3$ .

$\int z^{- \frac{1}{3}} - 2 z^{\frac{3 \cdot 0}{3 \cdot 1}} d z$

Step 5.6.2.2

Cancel the common factor.

$\int z^{- \frac{1}{3}} - 2 z^{\frac{3 \cdot 0}{3 \cdot 1}} d z$

Step 5.6.2.3

Rewrite the expression.

$\int z^{- \frac{1}{3}} - 2 z^{\frac{0}{1}} d z$

Step 5.6.2.4

Divide $0$ by $1$ .

$\int z^{- \frac{1}{3}} - 2 z^{0} d z$

Step 5.7

Anything raised to $0$ is $1$ .

$\int z^{- \frac{1}{3}} - 2 \cdot 1 d z$

Step 5.8

Multiply $- 2$ by $1$ .

$\int z^{- \frac{1}{3}} - 2 d z$

Step 6

Split the single integral into multiple integrals.

$\int z^{- \frac{1}{3}} d z + \int - 2 d z$

Step 7

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

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

Step 8

Apply the constant rule.

$\frac{3}{2} z^{\frac{2}{3}} + C - 2 z + C$

Step 9

Simplify.

$\frac{3}{2} z^{\frac{2}{3}} - 2 z + C$