Integrating by Parts: Solve e^(-x)cos x dx

[Shaybay92]

Homework Statement

I have attempted and failed solving the following integration:

Integrate : e^(-x) cos x dx

Homework Equations

I tried using the integration by parts rule:

uv - (integral) v (du/dx) dx

The Attempt at a Solution

I let u = e^(-x) and dv/dx = cos x

therefore (du/dx) = -e^(-x) and v = sin x

e^(-x)sinx - (integral)-e^(-x)sinx dx

This does not seem to cancel out anything and just keeps cycling through e and sin/cos

 

[diazona]

Integrate by parts twice, and you will come up with an equation you can solve for the integral you want.

 

[Cyosis]

Alternatively you could write the integral as $\text{Re}(\int e^{-x}e^{i x} dx )$.

 

[Shaybay92]

I just end up with having to integrate exactly what I began with!

After doing parts twice i get

-e^-xcosx - $$\int$$ e^-x cosx dx

 

[Cyosis]

Where did the first term, e^(-x)sinx, from the first partial integration go? Now define $I=\int e^{-x}\cos x dx$. You will then get an equation I= (some stuff)-I, we want to know I therefore solve for I!