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JEE Main202622 January 2026Morning ShiftMathematicsIndefinite IntegrationActual

If ( x)^ -11 2 ( x)^ -5 2 d x=- p₁ q₁ ( x)^ 9 2 - p₂ q₂ ( x)^ 5 2 - p₃ q₃ ( x)^ 1 2 + p₄ q₄ ( 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 p₁ p₂ p₃ p₄ q₁ q₂ q₃ q₄ is equal to

Correct answer

0

Step-by-step solution

^ -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: p₁ q₁ = 2 9 , p₂ q₂ = 6 5 , p₃ q₃ = 6 1 , p₄ q₄ = 2 3 . 15p₁p₂p₃p₄ q₁q₂q₃q₄ = 15 2 6 6 2 9 5 1 3 = 2160 135 = 16 .

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