Calculus Examples

\int \frac{e^{\sqrt{y}}}{\sqrt{y}} d y

$\int \frac{e^{\sqrt{y}}}{\sqrt{y}} d y$

Step 1

Apply basic rules of exponents.

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Step 1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{y}

$\sqrt{y}$ as

y^{\frac{1}{2}}

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

\int \frac{e^{y^{\frac{1}{2}}}}{\sqrt{y}} d y

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

Step 1.2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{y}

$\sqrt{y}$ as

y^{\frac{1}{2}}

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

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

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

Step 1.3

Move

y^{\frac{1}{2}}

$y^{\frac{1}{2}}$ out of the denominator by raising it to the

- 1

$- 1$ power.

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

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

Step 1.4

Multiply the exponents in

{(y^{\frac{1}{2}})}^{- 1}

${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 1.4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.4.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

- 1

$- 1$ .

\int e^{y^{\frac{1}{2}}} y^{\frac{- 1}{2}} d y

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

Step 1.4.3

Move the negative in front of the fraction.

\int e^{y^{\frac{1}{2}}} y^{- \frac{1}{2}} d y

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

\int e^{y^{\frac{1}{2}}} y^{- \frac{1}{2}} d y

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

\int e^{y^{\frac{1}{2}}} y^{- \frac{1}{2}} d y

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

Step 2

Let

u = y^{\frac{1}{2}}

$u = y^{\frac{1}{2}}$ . Then

d u = \frac{1}{2 y^{\frac{1}{2}}} d y

$d u = \frac{1}{2 y^{\frac{1}{2}}} d y$ , so

- d u = \frac{- 1}{2 y^{\frac{1}{2}}} d y

$- d u = \frac{- 1}{2 y^{\frac{1}{2}}} d y$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 2.1

Let

u = y^{\frac{1}{2}}

$u = y^{\frac{1}{2}}$ . Find

\frac{d u}{d y}

$\frac{d u}{d y}$ .

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Step 2.1.1

Differentiate

y^{\frac{1}{2}}

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

\frac{d}{d y} [y^{\frac{1}{2}}]

$\frac{d}{d y} [y^{\frac{1}{2}}]$

Step 2.1.2

Differentiate using the Power Rule which states that

\frac{d}{d y} [y^{n}]

$\frac{d}{d y} [y^{n}]$ is

n y^{n - 1}

$n y^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

\frac{1}{2} y^{\frac{1}{2} - 1}

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

Step 2.1.3

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}}

$\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.1.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}

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

Step 2.1.5

Combine the numerators over the common denominator.

\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}}

$\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

\frac{1}{2} y^{\frac{1 - 2}{2}}

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

Step 2.1.6.2

Subtract

2

$2$ from

1

$1$ .

\frac{1}{2} y^{\frac{- 1}{2}}

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

Step 2.1.7

Move the negative in front of the fraction.

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

Step 2.1.8

Simplify.

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Step 2.1.8.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.8.2

Multiply

$\frac{1}{2}$ by

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

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

Step 2.2

Rewrite the problem using

$u$ and

$d u$ .

$\int 2 e^{u} d u$

Step 3

Since

$2$ is constant with respect to

$u$ , move

$2$ out of the integral.

$2 \int e^{u} d u$

Step 4

The integral of

$e^{u}$ with respect to

$u$ is

$e^{u}$ .

$2 (e^{u} + C)$

Step 5

Simplify.

$2 e^{u} + C$

Step 6

Replace all occurrences of

$u$ with

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

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