We can integrate by parts using the differential:
d(e^x) = e^xdxd(ex)=exdx
so that:
int e^xcosxdx = int cosx d(e^x) = e^xcosx - int e^xd(cosx)= e^xcosx + int e^xsinxdx∫excosxdx=∫cosxd(ex)=excosx−∫exd(cosx)=excosx+∫exsinxdx
We can solve this last integrate by parts again:
int e^xsinxdx = int sinx d(e^x) = e^xsinx - int e^xd(sinx) = e^xsinx - int e^xcosxdx∫exsinxdx=∫sinxd(ex)=exsinx−∫exd(sinx)=exsinx−∫excosxdx
so that we have:
int e^xcosxdx =e^xcosx + e^xsinx - int e^xcosxdx∫excosxdx=excosx+exsinx−∫excosxdx
Now the same integral is appearing at both members and we can solve for it:
2int e^xcosxdx =e^xcosx + e^xsinx2∫excosxdx=excosx+exsinx
int e^xcosxdx =e^x((cosx + sinx)/2)+C∫excosxdx=ex(cosx+sinx2)+C
