Answer:
s cos^(-1)(-(q^2 s)/2) + sqrt(4 - q^4 s^2)/q^2 + constant
Explanation:
Take the integral:
integral cos^(-1)(-(q^2 s)/2) ds
For the integrand cos^(-1)(-(q^2 s)/2), substitute u = -(q^2 s)/2 and du = -q^2/2 ds:
= -2/q^2 integral cos^(-1)(u) du
For the integrand cos^(-1)(u), integrate by parts, integral f dg = f g - integral g df, where
f = cos^(-1)(u), dg = du, df = -1/sqrt(1 - u^2) du, g = u:
= -(2 u cos^(-1)(u))/q^2 + 2/q^2 integral-u/sqrt(1 - u^2) du
Factor out constants:
= -(2 u cos^(-1)(u))/q^2 - 2/q^2 integral u/sqrt(1 - u^2) du
For the integrand u/sqrt(1 - u^2), substitute p = 1 - u^2 and dp = -2 u du:
= -(2 u cos^(-1)(u))/q^2 + 1/q^2 integral1/sqrt(p) dp
The integral of 1/sqrt(p) is 2 sqrt(p):
= (2 sqrt(p))/q^2 - (2 u cos^(-1)(u))/q^2 + constant
Substitute back for p = 1 - u^2:
= (2 sqrt(1 - u^2))/q^2 - (2 u cos^(-1)(u))/q^2 + constant
Substitute back for u = -(q^2 s)/2:
Answer: = s cos^(-1)(-(q^2 s)/2) + sqrt(4 - q^4 s^2)/q^2 + constant