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

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

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integral of (arctan(e^x))/(e^{2x)}

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

- \frac{1}{2} e^{- 2 x} arctan (e^{x}) - \frac{1}{2} (- arctan (e^{- x}) + e^{- x}) + C

Solution steps

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

Apply Integration By Parts

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

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

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

= - \frac{1}{2} e^{- 2 x} arctan (e^{x}) - \frac{1}{2} (- arctan (e^{- x}) + e^{- x})

Add a constant to the solution

= - \frac{1}{2} e^{- 2 x} arctan (e^{x}) - \frac{1}{2} (- arctan (e^{- x}) + e^{- x}) + C

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