What is the integral of (e^x)/(sqrt(e^{2x)-1)} ?

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

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integral of (e^x)/(sqrt(e^{2x)-1)}

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

ln e^{x} + e^{2 x} - 1 + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

= ln u^{2} + 1 + u

Substitute back

u = e^{2 x} - 1

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

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

= ln e^{x} + e^{2 x} - 1

Add a constant to the solution

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

\int_{0}^{4} \frac{3}{x ^{2}} d x

y = 10 - x^{2}, y = 4

\frac{d ^{2}}{d x ^{2}} (5 e^{x})

f (x) = 5 x^{2} - 4 x + 3

y^{^{''}} = k^{2} \cdot y

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