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

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

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

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

2 arcsin (e^{x}) + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 arcsin (u)

Substitute back

u = e^{x}

= 2 arcsin (e^{x})

Add a constant to the solution

= 2 arcsin (e^{x}) + C

9 x^{2} sin (x) tan (x)

\int a cos (t) d t

\frac{d y}{d x} = \frac{5 ( 3 - y ) ^{2}}{( 3 - x ) ^{2}}

\frac{d}{d x} (\frac{7}{10} x^{2} + 4^{3})

\frac{d y}{d x} = \frac{18 ( 2 + y ) ^{2}}{( 1 - 3 x ) ^{2}}

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