What is the integral of 2e^xsin(e^x-1) ?

The integral of 2e^xsin(e^x-1) is 2(e^xsin(e^x-1)+e^xsin(1-e^x)-cos(1-e^x))+C

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integral of 2e^xsin(e^x-1)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

= 2 (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})

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

Simplify

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

Add a constant to the solution

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

What is the integral of 2e^xsin(e^x-1) ?
The integral of 2e^xsin(e^x-1) is 2(e^xsin(e^x-1)+e^xsin(1-e^x)-cos(1-e^x))+C