What is the integral of e^xsin(e^x-2) ?

The integral of e^xsin(e^x-2) is e^xsin(e^x-2)+e^xsin(2-e^x)-cos(2-e^x)+C

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integral of e^xsin(e^x-2)

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

e^{x} sin (e^{x} - 2) + e^{x} sin (2 - e^{x}) - cos (2 - e^{x}) + C

Solution steps

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

Apply u-substitution

: \int sin (u - 2) d u

= \int sin (u - 2) d u

Apply Integration By Parts

: sin (u - 2) u - \int cos (u - 2) u d u

= sin (u - 2) u - \int cos (u - 2) u d u

\int cos (u - 2) u d u = - u sin (2 - u) + cos (2 - u)

= sin (u - 2) u - (- u sin (2 - u) + cos (2 - u))

Substitute back

u = e^{x}

= sin (e^{x} - 2) e^{x} - (- e^{x} sin (2 - e^{x}) + cos (2 - e^{x}))

Simplify

sin (e^{x} - 2) e^{x} - (- e^{x} sin (2 - e^{x}) + cos (2 - e^{x})) : e^{x} sin (e^{x} - 2) + e^{x} sin (2 - e^{x}) - cos (2 - e^{x})

= e^{x} sin (e^{x} - 2) + e^{x} sin (2 - e^{x}) - cos (2 - e^{x})

Add a constant to the solution

= e^{x} sin (e^{x} - 2) + e^{x} sin (2 - e^{x}) - cos (2 - e^{x}) + C

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What is the integral of e^xsin(e^x-2) ?
The integral of e^xsin(e^x-2) is e^xsin(e^x-2)+e^xsin(2-e^x)-cos(2-e^x)+C