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

Note by Fredirick Estrella 1 year, 8 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 \(y = \ln x \).

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TopNewestIn 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 \(y = \ln x \).

Log in to reply