If $$f(x)=limn$$→∞ xn−x−$$nxn+x$$−n,x>$$1$$ then ∫$$xf(x)log$$⁡$$x+1+x21+x2dx$$ is

Correct option is 4none of these

Questions

If $f (x) = lim_{n \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}}, x > 1$ then
$\int \frac{x f (x) \log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}}$ dx is

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log⁡x+1+x2−x+C

12x2log⁡x+1+x2−1+C

(x−1)log⁡x+1+x2+C

none of these

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detailed solution

Correct option is D

f′(x)=limn→∞ xn−x−nxn+x−n=limn→∞ 1−(1/x)2n1+(1/x)2n=1(1/x<1) Using integration by parts , we get I=∫xlog⁡x+1+x21+x2dx=1+x2log⁡x+1+x2dx−∫dx=1+x2log⁡x+1+x2−x+C.

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If f(x)=limn→∞ xn−x−nxn+x−n,x>1 then ∫xf(x)log⁡x+1+x21+x2dx is

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If $f (x) = lim_{n \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}}, x > 1$ then
$\int \frac{x f (x) \log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}}$ dx is