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JEE Main202529 Jan 2025Evening ShiftMathematicsLimitsActual

If _ t 0 ( ₀^1(3 x+5)^ t d x )^ 1 t = 5 e ( 8 5 )^ 2 3 , then is equal to ________

Correct answer

0

Step-by-step solution

aligned 5 e & = ( _ t 0 1 t ( ₀^1(3 x+5)^t d x-1 ) ) & = ( _ t 0 1 t ( (3 x+5)^ t+1 3(t+1) )₀^1-1 ) & = ( _ t 0 1 t ( 8^ t+1 -5^ t+1 3(t+1) -1 ) ) & = ( _ t 0 1 t ( 8^ t+1 -5^ t+1 -3 t-3 3(t+1) ) ) & = ( _ t 0 ( 8^ t+1 8-5^ t+1 5-3 3(t+1) ) ) & = ( 8^8- 5^5-3 5 ) aligned aligned & = ( 8 5 )^ 2 / 3 5 e = ( ( 8^8 5^5 ) 5 -1 ) & ( 8 5 )^ 2 / 3 5 = ( 8^8 5^5 )^ 1 / 3 = ( 8^6 8^2 5^3 5^2 )^ 1 / 3 = 64 5 ( 8 5 )^ 2 / 3 & =64 aligned

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