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For a continuous function $f$ , the value $\int_{0}^{\infty} f (x^{n} + x^{- n}) \frac{\log x}{x} + \frac{1}{1 + x^{2}} d x$ is

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Correct option is A

Putting 1x=t⇒dx=−1t2dt, soI=∫0∞ fxn+x−nlog⁡xdxx=−∫∞0 ft−n+tnlog⁡1ttdtt2=−∫0∞ ft−n+tnlog⁡tdtt=−I⇒ 2I=0⇒I=0The given integral is equal to ∫0∞ 11+x2dx=tan−1⁡x0∞=π2

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For a continuous function f, the value ∫0∞ fxn+x−nlog⁡xx+11+x2dx is

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For a continuous function $f$ , the value $\int_{0}^{\infty} f (x^{n} + x^{- n}) \frac{\log x}{x} + \frac{1}{1 + x^{2}} d x$ is