Calculus Examples

$\int \frac{1}{y \sqrt{y}} d y$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify the expression.

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Step 2.1

Simplify.

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

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

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Combine the numerators over the common denominator.

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

Step 2.1.2.4

Add $2$ and $1$ .

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Move $y^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2.2

Multiply the exponents in ${(y^{\frac{3}{2}})}^{- 1}$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 2.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 3

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

$- 2 y^{- \frac{1}{2}} + C$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $- 2 y^{- \frac{1}{2}} + C$ as $- 2 \frac{1}{y^{\frac{1}{2}}} + C$ .

$- 2 \frac{1}{y^{\frac{1}{2}}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $- 2$ and $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{- 2}{y^{\frac{1}{2}}} + C$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{2}{y^{\frac{1}{2}}} + C$