Calculus Examples

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

Step 1

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

$\frac{d}{d x} [\int \frac{1}{t \sqrt{2 t}} \cdot \frac{\sqrt{2 t}}{\sqrt{2 t}} d t d t]$

Step 2

Combine and simplify the denominator.

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Step 2.1

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t \sqrt{2 t} \sqrt{2 t}} d t d t]$

Step 2.2

Move $\sqrt{2 t}$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t (\sqrt{2 t} \sqrt{2 t})} d t d t]$

Step 2.3

Raise $\sqrt{2 t}$ to the power of $1$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t ({\sqrt{2 t}}^{1} \sqrt{2 t})} d t d t]$

Step 2.4

Raise $\sqrt{2 t}$ to the power of $1$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t ({\sqrt{2 t}}^{1} {\sqrt{2 t}}^{1})} d t d t]$

Step 2.5

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {\sqrt{2 t}}^{1 + 1}} d t d t]$

Step 2.6

Add $1$ and $1$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {\sqrt{2 t}}^{2}} d t d t]$

Step 2.7

Rewrite ${\sqrt{2 t}}^{2}$ as $2 t$ .

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Step 2.7.1

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {({(2 t)}^{\frac{1}{2}})}^{2}} d t d t]$

Step 2.7.2

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {(2 t)}^{\frac{1}{2} \cdot 2}} d t d t]$

Step 2.7.3

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {(2 t)}^{\frac{2}{2}}} d t d t]$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {(2 t)}^{\frac{2}{2}}} d t d t]$

Step 2.7.4.2

Rewrite the expression.

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t {(2 t)}^{1}} d t d t]$

Step 2.7.5

Simplify.

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t (2 t)} d t d t]$

Step 3

Multiply $t$ by $t$ by adding the exponents.

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Step 3.1

Move $t$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t \cdot t \cdot 2} d t d t]$

Step 3.2

Multiply $t$ by $t$ .

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{t^{2} \cdot 2} d t d t]$

Step 4

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

$\frac{d}{d x} [\int \frac{\sqrt{2 t}}{2 t^{2}} d t d t]$

Step 5

Since $\int \frac{\sqrt{2 t}}{2 t^{2}} d t d t$ is constant with respect to $x$ , the derivative of $\int \frac{\sqrt{2 t}}{2 t^{2}} d t d t$ with respect to $x$ is $0$ .

$0$