Why does 1/x integrate to lnx?
If we let y = lnx, we then know that x = ey. By differentiating both sides of this equation with respect to y we get:dx/dy = ey, as the exponential function differentiates to itself when differentiated with respect to its power.But, as we noted earlier, x = ey, so we can substitute this in to get dx/dy = x.We can then take reciprocals of both sides to get dy/dx = 1/x.In other words the derivative of lnx is 1/x.But we know that integration is the opposite of differentiation (Fundamental Theorem of Calculus), giving us:The integral of 1/x is lnx.
