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

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

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

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

2 arctan (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 = arctan (u)

= 2 arctan (u)

Substitute back

u = e^{x}

= 2 arctan (e^{x})

Add a constant to the solution

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

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

\int x e^{x} x d x

\int (3 - 4 x)^{11} d x

\int \frac{1}{( 1 - 4 x ^{2} )} d x

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

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