Let's pick u, dvu,dv and solve for v, du:v,du:
u=e^-xu=e−x
du=-e^-xdxdu=−e−xdx
dv=cosxdxdv=cosxdx
v=sinxv=sinx
Thus, we apply intudv=uv-intvdu∫udv=uv−∫vdu:
inte^-xcosxdx=e^-xsinx+inte^-xsinxdx∫e−xcosxdx=e−xsinx+∫e−xsinxdx
This didn't yield anything solvable, let's integrate by parts once more, for inte^-xsinxdx∫e−xsinxdx:
u=e^-xu=e−x
du=-e^-xdxdu=−e−xdx
dv=sinxdxdv=sinxdx
v=-cosxv=−cosx
Applying the integration by parts formula, we get
inte^-xsinxdx=-e^-xcosx-inte^-xcosxdx∫e−xsinxdx=−e−xcosx−∫e−xcosxdx
There's nothing here we can solve, but note how our original integral showed up again.
Now, we said
inte^-xcosxdx=e^-xsinx+inte^-xsinxdx∫e−xcosxdx=e−xsinx+∫e−xsinxdx
So, let's plug in what we got for inte^-xsinxdx∫e−xsinxdx in:
inte^-xcosxdx=e^-xsinx-e^-xcosx-inte^-xcosxdx∫e−xcosxdx=e−xsinx−e−xcosx−∫e−xcosxdx
Solve for inte^-xcosxdx:∫e−xcosxdx:
2inte^-xcosxdx=e^-xsinx-e^-xcosx2∫e−xcosxdx=e−xsinx−e−xcosx
inte^-xcosxdx=1/2(e^-xsinx-e^-xcosx)∫e−xcosxdx=12(e−xsinx−e−xcosx)
Note I have not put in a constant of integration because we're going to be taking a definite integral:
int_0^2e^-xcosxdx=1/2(e^-xsinx-e^-xcosx)|_0^2=1/2(e^-2sin2-e^-2cos2+1)∫20e−xcosxdx=12(e−xsinx−e−xcosx)∣20=12(e−2sin2−e−2cos2+1)
