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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
9 months, 2 weeks 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 \(y = \ln x \). Pi Han Goh · 9 months, 2 weeks ago

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