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If ( x) -112( x) -52 dx = - p1q1( x) 92 - p2q2( x) 52 - p3q3( x) 12 + p4q4( x) -32 + C, where pi and qi are positive integers with (pi, qi) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4 is equal to
Answer
16
Solution
Sum of the powers = - 112 - 52 = -8, so divide and multiply by -8 x:
I = ( x)- 112 8 x dx = ( x)- 112(1 + 2 x)3 2 x dx
Put x = t, 2 x dx = dt:
I = t- 112(1 + 3t2 + 3t4 + t6) dt
= (t- 112 + 3t- 72 + 3t- 32 + t 12 )dt
= - 29t- 92 - 65t- 52 - 6t- 12 + 23t 32 + C
With t-1 = x:
I = - 29( x) 92 - 65( x) 52 - 61( x) 12 + 23( x) -32 + C
p1q1 = 29, p2q2 = 65, p3q3 = 61, p4q4 = 23
15p1p2p3p4q1q2q3q4 = 15 2 6 6 29 5 1 3 = 2160135 = 16
Answer: 16
JEE Main 2026 (Online) 22nd January Morning Shift