`int e^x cos(e^x) dx` Evaluate the indefinite integral.

You need to use the following substitution  `e^x= t` , such that:

`e^x = t=>e^x dx = dt `

`int e^x*cos(e^x) dx = int cos t dt`

`int cos t dt = sin t + c`

Replacing back `e^x` for `t` yields:

`int e^x*cos(e^x) dx = sin(e^x) + c`

Hence, evaluating the indefinite integral, yields `int e^x*cos(e^x) dx = sin(e^x) + c.`