Computer Sciences > GATE 2025 SET-2 > Calculus
The value of x such that x>1 satisfying the equation ∫1xt ln t dt=1/4 is
Correct : a
The correct answer is Option A - √e.
We need to find x > 1 such that ∫1x t ln t dt = 1/4.
Evaluating the integral using integration by parts:
Let u = ln t → du = (1/t) dt
Let dv = t dt → v = t2/2
∫t ln t dt = (t2/2) ln t − ∫(t2/2)(1/t) dt = (t2/2) ln t − t2/4 + C
Applying limits from 1 to x:
[(t2/2) ln t − t2/4]1x
= (x2/2) ln x − x2/4 − [(1/2) ln 1 − 1/4]
= (x2/2) ln x − x2/4 − [0 − 1/4]
= (x2/2) ln x − x2/4 + 1/4
Set equal to 1/4 and solve:
(x2/2) ln x − x2/4 + 1/4 = 1/4
(x2/2) ln x = x2/4
ln x = 1/2
x = e1/2 = √e
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