Quick question for Finding EOM with diff eq

[JaDi13]

I have been going through my old books again, and found myself a little stuck. I am not entirely sure if this would be better in this one or diffy eq.
The problem starts with having you find equation of motions when a= -bv, where b is constant and v = v(t)

Using method of separable equations, I was able to integrate the acceleration to get the velocity, v = v₀e^-bt

The problem is when I try to find the equation for x.
When I integrate v, using u substitution, I get x = (v₀/b)e^-bt+x₀
The books says the answer x = x₀ + (v₀/b)(1-e^-bt)
Also when I take the derivative of the book's answer, I get what I found for v

Am I way off base? Did I miss a step?

 

[Ssnow]

I get ##x = (v_{0}/b)e^{-bt}+x_{0}##

because this is wrong ... , the integral ##x=\int_{0}^{t}v_{0}e^{-bt}dt+x_{0}\not=\frac{v_{0}}{b}e^{-bt}+x_{0}##
control the signs! (and also the definite integral ...)
Ssnow