"Calculate the integrals. ∫(ln⁡x)ln⁡x(1x+ln⁡(ln⁡x)x)dx" was answered by our experts

Rivka Thorpe

Answered question

2021-10-21

Calculate the integrals.

\int {(\ln x)}^{\ln x} (\frac{1}{x} + \frac{\ln (\ln x)}{x}) d x

Elberte

Skilled2021-10-22Added 95 answers

The given integration can be written as:

\int {(\ln x)}^{\ln x} (\frac{1}{x} + \frac{\ln (\ln x)}{x}) d x = \int (\frac{1}{x} {\ln (x)}^{\ln (x)} + \frac{\ln (\ln (x))}{x} {\ln (x)}^{\ln (x)})

Put

u = \ln (x)

\int (\frac{1}{x} {\ln (x)}^{\ln (x)}) d x, d u = \frac{1}{x} d x

\int (\frac{1}{x} {\ln (x)}^{\ln (x)}) d x = \int u^{2} d u = \frac{u^{3}}{3} = \frac{1}{3} \ln^{3} (x)

Now solve the second integral:
put

u = \ln (x)

\int (\frac{\ln (\ln (x))}{x} {\ln (x)}^{\ln (x)}) d x

\int \ln (x^{\ln (x)}) \frac{\ln (\ln (x))}{x} d x = \int \ln (x) (\ln (x)) \frac{\ln (\ln (x))}{x} d x

= \int u^{2} \ln (u) d u

= \frac{1}{3} u^{3} \ln (u) - \int \frac{u^{2}}{3} d u

= \frac{1}{3} \ln^{3} (x) \ln (\ln (x)) - \frac{\ln^{3} (x)}{9}

Add a constant to the solution:

= \frac{1}{}

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