For the overall work of displacement from x=0 to the final extension l, w= -1/2kfl^2, evaluate the work of extension if kf(x)=a-b(x^(1/2)?

w = -∫₀ᴵ k x dx
w = -∫₀ᴵ (a − bx^1/2) x dx
w = -∫₀ᴵ (ax − bx^3/2) dx
w = -(a/2 x^2 − 2b/5 x^5/2) |₀ᴵ
w = -(a/2 I^2 − 2b/5 I^5/2)
w = -a/2 I^2 + 2b/5 I^5/2...Show more

I guess you are using "kf" to denote a single quantity, the force per unit length required for extending the spring (or whatever it is).

If kf(x) = a - b[x^(1/2)], then
w = integral from 0 to l of [a - bx^(1/2)] x dx
= integral from x = 0 to l of [ax - bx^(3/2)] dx
= (1/2)ax^2 - (2/5)bx^(5/2) to be evaluated at x = l and 0
= (1/2)al^2 - (2/5)bl^(5/2)....Show more