Electric Potential: Solving for Distance d from Origin

[funkwort]

A rod of length L lies along the x-axis with its left end at the origin and has a nonuniform charge density @ = &x. What is the electic potential a distance d from the origin?

V = k int dQ/x where:

dQ = &x dx so:

V = k& int (x/x+d)dx from (0 to L)
I let u = x+d and x = u-d
so:
V = k& int (1 - d/u)du = (x+d) - dln(x+d) (0 to L)
and I get: V = k&[L + dln(1 + d/L)]

in the book the answer is: k&[L - dln(1 + L/d)]

what am I doing wrong?

 

[Crosson]

look very carefully:

+ dln(1 + d/L) [your solution] = - dln(1 + L/d) [book's solution]

by the properties of logarithms.

 

[funkwort]

I know the properties of logarithms, but look:

(x+d) - dln(x+d) from (0 to L)

[(L+d) - dln(L+d)] - [d - dln(d)] = [L - dln(L+d) + dln(d)] = L + dln(d/L+d): so somewhere I must be making a mistake with the signs, but where?