What is the integral of e^x(x+sin(e^x)) ?

The integral of e^x(x+sin(e^x)) is e^xx-e^x-cos(e^x)+C

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integral of e^x(x+sin(e^x))

\int e^{x} (x + sin (e^{x})) d x

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

Solution steps

\int e^{x} (x + sin (e^{x})) d x

Apply u-substitution

= \int ln (u) + sin (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (u) d u + \int sin (u) d u

\int ln (u) d u = u ln (u) - u

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

= u ln (u) - u - cos (u)

Substitute back

u = e^{x}

= e^{x} ln (e^{x}) - e^{x} - cos (e^{x})

Simplify

ln (e^{x}) : x

= e^{x} x - e^{x} - cos (e^{x})

Add a constant to the solution

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

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