(I will assume logxlogx is natural log. (If not, insert ln10 where needed.)
int cos(log(x))dx = int xcos(log(x))*1/xdx∫cos(log(x))dx=∫xcos(log(x))⋅1xdx
Let u=xu=x and dvdv is the rest of the integrand.
Now we can integrate v = int cos(log(x))*1/xdx = sin(log(x))v=∫cos(log(x))⋅1xdx=sin(log(x))
(Use substitution with w=log(x)w=log(x))
Parts gives us:
int cos(log(x))dx = xsin(log(x)) - int sin(log(x)) dx∫cos(log(x))dx=xsin(log(x))−∫sin(log(x))dx
Do the same trick again to get
int cos(log(x))dx = xsin(log(x)) + xcos(log(x)) - int cos(log(x)) dx∫cos(log(x))dx=xsin(log(x))+xcos(log(x))−∫cos(log(x))dx
So
int cos(log(x))dx = 1/2(xsin(log(x)) + xcos(log(x))) ∫cos(log(x))dx=12(xsin(log(x))+xcos(log(x)))
