Question

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.

Answer 1

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