What is the integral of e^xarccos(e^x) ?

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

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integral of e^xarccos(e^x)

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

e^{x} arccos (e^{x}) - 1 - e^{2 x} + C

Solution steps

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

Apply Integration By Parts

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

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

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

= e^{x} arccos (e^{x}) - 1 - e^{2 x}

Add a constant to the solution

= e^{x} arccos (e^{x}) - 1 - e^{2 x} + C

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