What is the integral of e^xsin(e^x-1) ?

The integral of e^xsin(e^x-1) is e^xsin(e^x-1)+e^xsin(1-e^x)-cos(1-e^x)+C

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integral of e^xsin(e^x-1)

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

e^{x} sin (e^{x} - 1) + e^{x} sin (1 - e^{x}) - cos (1 - e^{x}) + C

Solution steps

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

Apply Integration By Parts

: e^{x} sin (e^{x} - 1) - \int e^{2 x} cos (e^{x} - 1) d x

= e^{x} sin (e^{x} - 1) - \int e^{2 x} cos (e^{x} - 1) d x

\int e^{2 x} cos (e^{x} - 1) d x = - e^{x} sin (1 - e^{x}) + cos (1 - e^{x})

= e^{x} sin (e^{x} - 1) - (- e^{x} sin (1 - e^{x}) + cos (1 - e^{x}))

Simplify

= e^{x} sin (e^{x} - 1) + e^{x} sin (1 - e^{x}) - cos (1 - e^{x})

Add a constant to the solution

= e^{x} sin (e^{x} - 1) + e^{x} sin (1 - e^{x}) - cos (1 - e^{x}) + C

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What is the integral of e^xsin(e^x-1) ?
The integral of e^xsin(e^x-1) is e^xsin(e^x-1)+e^xsin(1-e^x)-cos(1-e^x)+C