What is the integral of 1/(sqrt(e^x+2)) ?

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

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integral of 1/(sqrt(e^x+2))

\int \frac{1}{e ^{x} + 2} d x

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

Solution steps

\int \frac{1}{e ^{x} + 2} d x

Apply u-substitution

: \int \frac{1}{u u + 2} d u

= \int \frac{1}{u u + 2} d u

Apply u-substitution

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

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

Simplify

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

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

Add a constant to the solution

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

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