Calculus Examples

$\int_{0}^{\ln (x)} \frac{1}{\sqrt{4 + e^{t}}} d t$

Step 1

Multiply $\frac{1}{\sqrt{4 + e^{t}}}$ by $\frac{\sqrt{4 + e^{t}}}{\sqrt{4 + e^{t}}}$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{1}{\sqrt{4 + e^{t}}} \cdot \frac{\sqrt{4 + e^{t}}}{\sqrt{4 + e^{t}}} d t]$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt{4 + e^{t}}}$ by $\frac{\sqrt{4 + e^{t}}}{\sqrt{4 + e^{t}}}$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{\sqrt{4 + e^{t}} \sqrt{4 + e^{t}}} d t]$

Step 2.2

Raise $\sqrt{4 + e^{t}}$ to the power of $1$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{\sqrt{4 + e^{t}}}^{1} \sqrt{4 + e^{t}}} d t]$

Step 2.3

Raise $\sqrt{4 + e^{t}}$ to the power of $1$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{\sqrt{4 + e^{t}}}^{1} {\sqrt{4 + e^{t}}}^{1}} d t]$

Step 2.4

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

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{\sqrt{4 + e^{t}}}^{1 + 1}} d t]$

Step 2.5

Add $1$ and $1$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{\sqrt{4 + e^{t}}}^{2}} d t]$

Step 2.6

Rewrite ${\sqrt{4 + e^{t}}}^{2}$ as $4 + e^{t}$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + e^{t}}$ as ${(4 + e^{t})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{({(4 + e^{t})}^{\frac{1}{2}})}^{2}} d t]$

Step 2.6.2

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

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{(4 + e^{t})}^{\frac{1}{2} \cdot 2}} d t]$

Step 2.6.3

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

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{(4 + e^{t})}^{\frac{2}{2}}} d t]$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{(4 + e^{t})}^{\frac{2}{2}}} d t]$

Step 2.6.4.2

Rewrite the expression.

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{{(4 + e^{t})}^{1}} d t]$

Step 2.6.5

Simplify.

$\frac{d}{d x} [\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{4 + e^{t}} d t]$

Step 3

Take the derivative of $\int_{0}^{\ln (x)} \frac{\sqrt{4 + e^{t}}}{4 + e^{t}} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\ln (x)] \frac{\sqrt{4 + e^{\ln (x)}}}{4 + e^{\ln (x)}}$

Step 4

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x} \cdot \frac{\sqrt{4 + e^{\ln (x)}}}{4 + e^{\ln (x)}}$

Step 5

Exponentiation and log are inverse functions.

$\frac{1}{x} \cdot \frac{\sqrt{4 + x}}{4 + e^{\ln (x)}}$

Step 6

Exponentiation and log are inverse functions.

$\frac{1}{x} \cdot \frac{\sqrt{4 + x}}{4 + x}$

Step 7

Multiply $\frac{1}{x}$ by $\frac{\sqrt{4 + x}}{4 + x}$ .

$\frac{\sqrt{4 + x}}{x (4 + x)}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{\sqrt{4 + x}}{x \cdot 4 + x \cdot x}$

Step 8.2

Combine terms.

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Step 8.2.1

Move $4$ to the left of $x$ .

$\frac{\sqrt{4 + x}}{4 \cdot x + x \cdot x}$

Step 8.2.2

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

$\frac{\sqrt{4 + x}}{4 x + x^{1} x}$

Step 8.2.3

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

$\frac{\sqrt{4 + x}}{4 x + x^{1} x^{1}}$

Step 8.2.4

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

$\frac{\sqrt{4 + x}}{4 x + x^{1 + 1}}$

Step 8.2.5

Add $1$ and $1$ .

$\frac{\sqrt{4 + x}}{4 x + x^{2}}$

Step 8.3

Reorder terms.

$\frac{\sqrt{4 + x}}{x^{2} + 4 x}$

Step 8.4

Factor $x$ out of $x^{2} + 4 x$ .

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Step 8.4.1

Factor $x$ out of $x^{2}$ .

$\frac{\sqrt{4 + x}}{x \cdot x + 4 x}$

Step 8.4.2

Factor $x$ out of $4 x$ .

$\frac{\sqrt{4 + x}}{x \cdot x + x \cdot 4}$

Step 8.4.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{\sqrt{4 + x}}{x (x + 4)}$