Suppose f is a function of x.
1) find d/dx f^2
2)use the result to find the integral ∫ lnx/xdx.
I did 1), and I got 2fdf, but im confused about number 2.
d/dx f^2 = 2f df/dx
∫ lnx/x dx
Since we are discussing f^2, let's let
f(x) = (lnx)^2
df/dx = 2 lnx * 1/x = 2lnx/x
So, clearly,
∫lnx/x dx = (lnx)^2/2 + C
check, using integration by parts:
u = lnx
dv = dx/x
du = 1/x dx
v = lnx
∫lnx/x dx = lnx*lnx - ∫ lnx/x dx
2∫lnx/x dx = (lnx)^2
∫lnx/xdx = (lnx)^2/2 + C