What is the integral of (e^x)/(3-e^{2x)} ?

The integral of (e^x)/(3-e^{2x)} is 1/(2sqrt(3))(ln|(e^x)/(sqrt(3))+1|-ln|(e^x)/(sqrt(3))-1|)+C

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integral of (e^x)/(3-e^{2x)}

\int \frac{e ^{x}}{3 - e ^{2 x}} d x

\frac{1}{2 3} (ln \frac{e ^{x}}{3} + 1 - ln \frac{e ^{x}}{3} - 1) + C

Solution steps

\int \frac{e ^{x}}{3 - e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{3 - u ^{2}} d u

= \int \frac{1}{3 - u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{3 ( 1 - v ^{2} )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{1 - v ^{2}} d v

Use the common integral:

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

= \frac{1}{3} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= \frac{1}{3} \frac{ln \frac{e ^{x}}{3} + 1}{2} - \frac{ln \frac{e ^{x}}{3} - 1}{2}

Simplify

\frac{1}{3} \frac{ln \frac{e ^{x}}{3} + 1}{2} - \frac{ln \frac{e ^{x}}{3} - 1}{2} : \frac{1}{2 3} (ln \frac{e ^{x}}{3} + 1 - ln \frac{e ^{x}}{3} - 1)

= \frac{1}{2 3} (ln \frac{e ^{x}}{3} + 1 - ln \frac{e ^{x}}{3} - 1)

Add a constant to the solution

= \frac{1}{2 3} (ln \frac{e ^{x}}{3} + 1 - ln \frac{e ^{x}}{3} - 1) + C

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x \to 1.7 lim (sin (x))

\int - \frac{19}{x ^{2} + 1} d x

What is the integral of (e^x)/(3-e^{2x)} ?
The integral of (e^x)/(3-e^{2x)} is 1/(2sqrt(3))(ln|(e^x)/(sqrt(3))+1|-ln|(e^x)/(sqrt(3))-1|)+C