Calculus Examples

$\int y^{2} \sqrt[3]{y} d y$

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

$\int y^{2} y^{\frac{1}{3}} d y$

Step 1.2

Multiply $y^{2}$ by $y^{\frac{1}{3}}$ by adding the exponents.

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Step 1.2.1

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

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

Step 1.2.2

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

$\int y^{2 \cdot \frac{3}{3} + \frac{1}{3}} d y$

Step 1.2.3

Combine $2$ and $\frac{3}{3}$ .

$\int y^{\frac{2 \cdot 3}{3} + \frac{1}{3}} d y$

Step 1.2.4

Combine the numerators over the common denominator.

$\int y^{\frac{2 \cdot 3 + 1}{3}} d y$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $2$ by $3$ .

$\int y^{\frac{6 + 1}{3}} d y$

Step 1.2.5.2

Add $6$ and $1$ .

$\int y^{\frac{7}{3}} d y$

Step 2

By the Power Rule, the integral of $y^{\frac{7}{3}}$ with respect to $y$ is $\frac{3}{10} y^{\frac{10}{3}}$ .

$\frac{3}{10} y^{\frac{10}{3}} + C$