Calculus Examples

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

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

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

$x \cdot x + x \cdot 4$ .

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