Calculus Examples

$\int y^{2} \sqrt{y} d y$

Step 1

Simplify.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 1.2

Multiply $y^{2}$ by $y^{\frac{1}{2}}$ 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.

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $2$ by $2$ .

$\int y^{\frac{4 + 1}{2}} d y$

Step 1.2.5.2

Add $4$ and $1$ .

$\int y^{\frac{5}{2}} d y$

Step 2

By the Power Rule, the integral of $y^{\frac{5}{2}}$ with respect to $y$ is $\frac{2}{7} y^{\frac{7}{2}}$ .

$\frac{2}{7} y^{\frac{7}{2}} + C$