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# Natural logarithm of natural logarithm

Can you find the integral of $$\int { \ln { \left( \ln { x } \right) } dx }$$

Note by Fredirick Estrella
11 months ago

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In terms of elementary functions? No.

But you can include the logarithmic integral $$\displaystyle \text{li}(x) = \int_0^x \dfrac{dt}{\ln t}$$, to get $$\displaystyle \int \ln(\ln x) \, dx = x \ln(\ln x) - \text{li}(x) + C$$.

Your first step is to use the substitution $$y = \ln x$$. · 11 months ago