The antiderivative of $$f(x)=log$$⁡(log$$⁡$$x)+(log$$⁡$$x)−$$2$$ whose graph passes through (e$$,$$e) is

Question

The antiderivative of $$f(x)=log$$⁡(log$$⁡$$x)+(log$$⁡$$x)−$$2$$ whose graph passes through (e$$,$$e) is

Infinity Learn · Accepted Answer

An antiderivative of f (x) $$=$$ $$F(x)$$ $$=$$∫log⁡(log$$⁡$$x)+(log$$⁡$$x)−$$2dx+C=xlog$$⁡(log$$⁡$$x)−∫xxlog⁡$$xdx+$$∫(log$$⁡$$x)−$$2dx+C$$ (integrating$$ by parts the first $$term) $$=xlog$$⁡(log$$⁡$$x)−$$x(log$$⁡$$x)$$−$$1+$$∫(log$$⁡$$x)−$$2dx+$$∫(log$$⁡$$x)−$$2dx+C$$ (again$$ integrating by parts $$(log$$⁡$$x)−$$1$$ $$=xlog$$⁡(log$$⁡$$x)−$$x(log$$⁡$$x)$$−$$1+CPutting$$ $$x=e$$ we have $$e=0$$−$$e+C$$ so $$C=2eThus$$ $$F(x)=xlog$$⁡(log$$⁡$$x)−(log$$⁡$$x)−$$1+2e$$

The antiderivative of $f (x) = \log (\log x) +$ $(\log x)^{- 2}$
whose graph passes through $(e, e)$ is

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The antiderivative of f(x)=log⁡(log⁡x)+(log⁡x)−2 whose graph passes through (e,e) is

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The antiderivative of $f (x) = \log (\log x) +$ $(\log x)^{- 2}$
whose graph passes through $(e, e)$ is