GATE 2020 MA – Question 63
For $F(x)=\int_1^x[\log t/(t-1)]dt$, 0 < x < 2, find the coefficient of (x−1)⁵ in its Taylor expansion about x = 1, to two decimal places.
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Correct answer: 0.04 to 0.04
Explanation
Write h = t − 1. The removable integrand expands as $\log(1+h)/h=1-h/2+h^2/3-h^3/4+h^4/5-\cdots$. Integrating from 0 to x − 1 gives coefficients $h-h^2/4+h^3/9-h^4/16+h^5/25-\cdots$. Thus the fifth-power coefficient is 1/25 = 0.04.