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The value of $\int_{0}^{1} \frac{x^{2 α} - 1}{\log x} d x, if α = \frac{2 n - 1}{2}$ is

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log n

2 log n

log 2n

(1/2) log n

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detailed solution

Correct option is C

Let I=∫01 x2α−1log⁡xdx⇒dIdα=∫01 2x2αlog⁡xlog⁡xdx=22α+1x2α+101=22α+1⇒I=log⁡(2α+1)+CWhen α=0,I=0⇒C=0Thus I=log⁡(2α+1)=log⁡(2n)

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The value of ∫01 x2α−1log⁡x d x, if α=2n−12 is

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The value of $\int_{0}^{1} \frac{x^{2 α} - 1}{\log x} d x, if α = \frac{2 n - 1}{2}$ is