If ( x)^ -11 2 ( x)^ -5 2 d x= - p_ 1 q_ 1 ( x)^ 9 2 - p_ 2 q_ 2 ( x)^ 5 2 - p_ 3 q_ 3 ( x)^ 1 2 + p_ 4 q_ 4 ( x)^ -3 2 + C , where p_ i and q_ i are positive integers with gcd (p_ i , q_ i )=1 for i=1,2,3,4 and C is the constant of integration, then 15 p_ 1 p_ 2 p_ 3 p_ 4 q_ 1 q_ 2 q_ 3 q_ 4 is equal to
A. A
^ -11/2 x ^ -5/2 x\,dx. Since m+n = -8 (negative even), substitute t = x, dx = -dt 1+t^2 . Integral becomes - (1+t^2)^3 t^ 5/2 dt = - (t^ -5/2 +3t^ -1/2 +3t^ 3/2 +t^ 7/2 )dt. = 2 3 t^ -3/2 - 6t^ 1/2 - 6 5 t^ 5/2 - 2 9 t^ 9/2 + C. Comparing: p_1 q_1 = 2 9 , p_2 q_2 = 6 5 , p_3 q_3 = 6 1 , p_4 q_4 = 2 3 . 15p_1p_2p_3p_4 q_1q_2q_3q_4 = 15 2 6 6 2 9 5 1 3 = 2160 135 = 16.
Related: Mathematics — Indefinite Integration · All PYQ Banks