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GATE CSE 2025 Session 2 Q12: Solve ∫₁ˣ t ln t dt = 1/4 for x > 1

Question

The value of xx such that x>1x > 1, satisfying the equation ∫1xtln⁡t dt=14\displaystyle\int_1^x t \ln t \, dt = \frac{1}{4} is

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Answer: A (official GATE 2025 final answer key)

Concept used: Integration by parts.

First step: ∫tln⁡t dt=t22ln⁡t−t24\int t \ln t \, dt = \frac{t^2}{2}\ln t - \frac{t^2}{4}; evaluate from 1 to xx and set it equal to 14\frac14.

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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