Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

Let $u = t^{\frac{1}{2}}$ . Then $d u = \frac{1}{2 t^{\frac{1}{2}}} d t$ , so $- d u = \frac{- 1}{2 t^{\frac{1}{2}}} d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = t^{\frac{1}{2}}$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $t^{\frac{1}{2}}$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 2.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.1.4

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

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

Step 2.1.5

Combine the numerators over the common denominator.

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

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $- 1$ by $2$ .

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

Step 2.1.6.2

Subtract $2$ from $1$ .

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

Step 2.1.7

Move the negative in front of the fraction.

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

Step 2.1.8

Simplify.

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Step 2.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.8.2

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

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

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

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