Calculus Examples

$\int y \sqrt[3]{y} d y$

Step 1

Simplify.

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Step 1.1

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

$\int y \cdot y^{\frac{1}{3}} d y$

Step 1.2

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

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Step 1.2.1

Multiply $y$ by $y^{\frac{1}{3}}$ .

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Step 1.2.1.1

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

$\int y^{1} y^{\frac{1}{3}} d y$

Step 1.2.1.2

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

$\int y^{1 + \frac{1}{3}} d y$

Step 1.2.2

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

$\int y^{\frac{3}{3} + \frac{1}{3}} d y$

Step 1.2.3

Combine the numerators over the common denominator.

$\int y^{\frac{3 + 1}{3}} d y$

Step 1.2.4

Add $3$ and $1$ .

$\int y^{\frac{4}{3}} d y$

Step 2

By the Power Rule, the integral of $y^{\frac{4}{3}}$ with respect to $y$ is $\frac{3}{7} y^{\frac{7}{3}}$ .

$\frac{3}{7} y^{\frac{7}{3}} + C$