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

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

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

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

- arctan (e^{x}) + C

Solution steps

\int \frac{- 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

= - \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 = arctan (u)

= - arctan (u)

Substitute back

u = e^{x}

= - arctan (e^{x})

Add a constant to the solution

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

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