Calculus of a Single Variable, Chapter 5, 5.7, Section 5.7, Problem 24

Make the substitution u = sqrt(4x^2 - 9), then du = (4x)/sqrt(4x^2 - 9) dx. Inversely, dx =sqrt(4x^2 - 9)/(4x) du = u/(4x) du and 4x^2 = u^2 + 9. The limits of integration become from sqrt(3) to 3sqrt(3).
The indefinite integral becomes
int u/(4 u x^2) du = int (du)/(u^2 + 9) = 1/3 arctan(u/3) + C,
where C is an arbitrary constant.
Thus the definite integral is 1/3 (arctan(sqrt(3)) - arctan(1/sqrt(3))) = 1/3 (pi/3 - pi/6) = pi/18.