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$Evaluate \int (\log (\log x) + \frac{1}{(\log x)^{2}}) d x$

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a

xlog⁡(log⁡x)+x(log⁡x)−1+c

b

xlog⁡(log⁡x)−(log⁡x)−1+c

c

xlog⁡(log⁡x)−x(log⁡x)−1+c

d

None of these

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detailed solution

Correct option is C

∫log⁡(log⁡x)+1(log⁡x)2dx=∫1⋅log⁡(log⁡x)dx+∫dx(log⁡x)2 (apply uv rule to the first integrand)=xlog⁡(log⁡x)−∫1xlog⁡xxdx+∫dx(log⁡x)2+c=xlog⁡(log⁡x)−∫(log⁡x)−1dx+∫dx(log⁡x)2+c=xlog⁡(log⁡x)−x(log⁡x)−1+∫(log⁡x)−2dx+∫(log⁡x)−2dx=xlog⁡(log⁡x)−x(log⁡x)−1+c

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