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

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

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

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

arcsin (e^{x}) + C

Solution steps

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

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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{1 - u ^{2}} d u = arcsin (u)

= arcsin (u)

Substitute back

u = e^{x}

= arcsin (e^{x})

Add a constant to the solution

= arcsin (e^{x}) + C

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