Calculus Examples

$\frac{2 d t}{\sqrt{t}}$

Step 1

Since $\frac{2 t}{\sqrt{t}}$ is constant with respect to $d$ , move $\frac{2 t}{\sqrt{t}}$ out of the integral.

$\frac{2 t}{\sqrt{t}} \int d d \cdot d$

Step 2

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

$\frac{2 t}{\sqrt{t}} (\frac{1}{2} d^{2} + C)$

Step 3

Simplify the answer.

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Step 3.1

Rewrite $\frac{2 t}{\sqrt{t}} (\frac{1}{2} d^{2} + C)$ as $\frac{2 t \sqrt{t}}{t} (\frac{1}{2} d^{2}) + C$ .

$\frac{2 t \sqrt{t}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{2 t \cdot t^{\frac{1}{2}}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2

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

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Step 3.2.2.1

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

$\frac{2 (t^{\frac{1}{2}} t)}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

$\frac{2 (t^{\frac{1}{2}} t^{1})}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2.2.2

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

$\frac{2 t^{\frac{1}{2} + 1}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2.3

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

$\frac{2 t^{\frac{1}{2} + \frac{2}{2}}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2.4

Combine the numerators over the common denominator.

$\frac{2 t^{\frac{1 + 2}{2}}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.2.5

Add $1$ and $2$ .

$\frac{2 t^{\frac{3}{2}}}{t} (\frac{1}{2} d^{2}) + C$

Step 3.2.3

Move $t^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 t^{\frac{3}{2}} t^{- 1} (\frac{1}{2} d^{2}) + C$

Step 3.2.4

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

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Step 3.2.4.1

Move $t^{- 1}$ .

$2 (t^{- 1} t^{\frac{3}{2}}) (\frac{1}{2} d^{2}) + C$

Step 3.2.4.2

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

$2 t^{- 1 + \frac{3}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.4.3

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

$2 t^{- 1 \cdot \frac{2}{2} + \frac{3}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.4.4

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

$2 t^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.4.5

Combine the numerators over the common denominator.

$2 t^{\frac{- 1 \cdot 2 + 3}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.4.6

Simplify the numerator.

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Step 3.2.4.6.1

Multiply $- 1$ by $2$ .

$2 t^{\frac{- 2 + 3}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.4.6.2

Add $- 2$ and $3$ .

$2 t^{\frac{1}{2}} (\frac{1}{2} d^{2}) + C$

Step 3.2.5

Combine $\frac{1}{2}$ and $d^{2}$ .

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

Step 3.2.6

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

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

Step 3.2.7

Combine $t^{\frac{1}{2}}$ and $\frac{2 d^{2}}{2}$ .

$\frac{t^{\frac{1}{2}} (2 d^{2})}{2} + C$

Step 3.2.8

Move $2$ to the left of $t^{\frac{1}{2}}$ .

$\frac{2 \cdot t^{\frac{1}{2}} d^{2}}{2} + C$

Step 3.2.9

Cancel the common factor.

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

Step 3.2.10

Divide $t^{\frac{1}{2}} d^{2}$ by $1$ .

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