What is the integral of 1/(sqrt(4+e^x)) ?

The integral of 1/(sqrt(4+e^x)) is -1/2 ln|(sqrt(4+e^x))/2+1|+1/2 ln|(sqrt(4+e^x))/2-1|+C

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integral of 1/(sqrt(4+e^x))

\int \frac{1}{4 + e ^{x}} d x

- \frac{1}{2} ln \frac{4 + e ^{x}}{2} + 1 + \frac{1}{2} ln \frac{4 + e ^{x}}{2} - 1 + C

Solution steps

\int \frac{1}{4 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u 4 + u} d u

= \int \frac{1}{u 4 + u} d u

Apply u-substitution

: \int \frac{2}{v ^{2} - 4} d v

= \int \frac{2}{v ^{2} - 4} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{v ^{2} - 4} d v

Apply Integral Substitution

= 2 \cdot \int \frac{1}{2 ( w ^{2} - 1 )} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} (- \int \frac{1}{- w ^{2} + 1} d w)

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 2 \cdot \frac{1}{2} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

= 2 \cdot \frac{1}{2} - \frac{ln \frac{4 + e ^{x}}{2} + 1}{2} - \frac{ln \frac{4 + e ^{x}}{2} - 1}{2}

Simplify

2 \cdot \frac{1}{2} - \frac{ln \frac{4 + e ^{x}}{2} + 1}{2} - \frac{ln \frac{4 + e ^{x}}{2} - 1}{2} : - \frac{1}{2} ln \frac{4 + e ^{x}}{2} + 1 + \frac{1}{2} ln \frac{4 + e ^{x}}{2} - 1

= - \frac{1}{2} ln \frac{4 + e ^{x}}{2} + 1 + \frac{1}{2} ln \frac{4 + e ^{x}}{2} - 1

Add a constant to the solution

= - \frac{1}{2} ln \frac{4 + e ^{x}}{2} + 1 + \frac{1}{2} ln \frac{4 + e ^{x}}{2} - 1 + C

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What is the integral of 1/(sqrt(4+e^x)) ?
The integral of 1/(sqrt(4+e^x)) is -1/2 ln|(sqrt(4+e^x))/2+1|+1/2 ln|(sqrt(4+e^x))/2-1|+C