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

The integral of 2/(e^x-e^{-x)} is ln(tanh(x/2))+C

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

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

ln (tanh (\frac{x}{2})) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 2 \cdot \int \frac{1}{2 sinh ( x )} d x

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}{sinh ( x )} d x

Use the Hyperbolic identity:

\frac{1}{sinh ( x )} = csch (x)

= 2 \cdot \frac{1}{2} \cdot \int csch (x) d x

Use the common integral:

\int csch (x) d x = ln (tanh (\frac{x}{2}))

= 2 \cdot \frac{1}{2} ln (tanh (\frac{x}{2}))

Simplify

2 \cdot \frac{1}{2} ln (tanh (\frac{x}{2})) : ln (tanh (\frac{x}{2}))

= ln (tanh (\frac{x}{2}))

Add a constant to the solution

= ln (tanh (\frac{x}{2})) + C

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