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

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

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

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

arcsin (e^{x}) + C

Solution steps

\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

\int_{- 4}^{0} (\frac{y}{4} + y + 5) d y

\int sin^{2} (3 x) cos^{2} (3 x) d x

\frac{( 2 x ^{2} + 16 x + x ^{3^{\frac{1}{2}}} + 5 )}{x ^{3}}

\frac{d ^{2} y}{d x ^{2}} - 2 a \frac{d y}{d x} + a^{2} y = 0

y = x^{4} + 9 x^{2} - x, (1, 9)

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