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

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

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integral of cos(e^x)e^x

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

sin (e^{x}) + C

Solution steps

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

Apply u-substitution

= \int cos (u) d u

Use the common integral:

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

= sin (u)

Substitute back

u = e^{x}

= sin (e^{x})

Add a constant to the solution

= sin (e^{x}) + C