`int e^(cos(t)) sin(t) dt` Evaluate the indefinite integral.

You need to use the following substitution  `cos t=u` , such that:

`cos t=u=>- sint dt = du => sin t dt= -du`

`int e^(cos t)*sint dt = - int e^u du = -e^u + c`

Replacing back  `cos t ` for `u` yields:

`int e^(cos t)*sint dt = -e^(cos t) + c`

Hence, evaluating the indefinite integral, yields `int e^(cos t)*sint dt = -e^(cos t) + c.`