What is the integral of e^x*arctan(e^x) ?

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

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integral of e^x*arctan(e^x)

\int e^{x} \cdot arctan (e^{x}) d x

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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