Question

How do integrate `int e^(cos)(t) (sin 2t) dt` between a = 0, `b = pi`?

Answer 1

Start with `int_0^pi e^cos(t)sin(2t) dt = 2int_0^pie^cos(t)sin(t)*cos(t)dt`

Now by part

`-2int-sin(t)e^cos(t)*cos(t) dt`

`u'=-sin(t)e^cos(t)`
`u=e^cos(t)`

`v = cos(t)`
`v' = -sin(t)`

You have : `-2([cos(t)*e^cos(t)]_0^pi - int_0^pi-sin(t)e^cos(t) dt)`

`=-2[cos(t)*e^cos(t)-e^cos(t)]_0^pi`

`=4/e`