Indefinite Integral with integration by parts

[Physics-UG]

Homework Statement

Evaluate ∫e^-θcos2θ dθ

Homework Equations

Integration by parts formula
∫udv = uv -∫vdu

The Attempt at a Solution

So in calc II we just started integration by parts and I'm doing one of the assignment problems. I know I need to do the integration by parts twice, but I've hit a loop, or so it seems, which I know isn't right. Maybe someone can see my faults?

I set:
u = cos(2θ) v = -e^-θ
du = -2sin(2θ) dθ dv = e^-θ dθ

So by the formula I got:
= [cos(2θ)] [-e^-θ] -2∫ [-e^-θ] [sin(2θ) dθ]

Here I used the second integration by parts:

u = sin2θ v = e^-θ
du = 2cos(2θ) dθ dv = -e^-θ dθ

Solving by the formula again:
= [sin(2θ)] [e^-θ] -2∫ [e^-θ] [cos(2θ) dθ]

I'm not too sure where I've made my algebraic error, or if I'm on the right track and this won't just put my into a loop giving the same equation above? First question I've posted, so hopefully it follows the format ok, if it doesn't, chime in and let me know so I can fix it properly. Thanks in advance everyone!

 

[phion]

Your substitutions are correct, but you're stopping a little short. The trick with integration by parts composed of an exponential function and a trigonometric function is to add to both sides the first integral you find that is a multiple of your original integral. From there, it should be obvious where to go next.

 

[Physics-UG]

Your substitutions are correct, but you're stopping a little short. The trick with integration by parts composed of an exponential function and a trigonometric function is to add to both sides the first integral you find that is a multiple of your original integral. From there, it should be obvious where to go next.

Ohh, hat's definitely what I'm missing! Thanks a bunch for the help!