Question

How do you integrate `int e^x sin x dx` using integration by parts?

Answer 1

`inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c`

Explanation:

Say `dv = e^x` so `v = e^x`, `u = sin(x)` so `du = cos(x)`

`inte^xsin(x)dx = e^xsin(x) - inte^(x)cos(x)dx`

Say `dv = e^x` so `v = e^x`, `u = cos(x)` so `du = -sin(x)`

`inte^xsin(x)dx = e^xsin(x) - (e^xcos(x) +inte^xsin(x)dx)`
`inte^xsin(x)dx = e^xsin(x) - e^xcos(x) -inte^xsin(x)dx`
`2inte^xsin(x)dx = e^xsin(x) - e^xcos(x)`
`inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c`