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Question Number 194412 by SaRahAli last updated on 05/Jul/23

Answered by som(math1967) last updated on 05/Jul/23

$$\int ln\left(lnx\right)dx+\int \frac{dx}{{\left(lnx\right)}^2}$$$$=ln\left(lnx\right)\int dx-\int \{\frac{d}{dx}ln\left(lnx\right)\int dx\}dx$$$$+\int \frac{dx}{{\left(lnx\right)}^2}$$$$=xln\left(lnx\right)-\int \frac{xdx}{xlnx}+\int \frac{dx}{{\left(lnx\right)}^2}$$$$=xln\left(lnx\right)-\frac{1}{lnx}\int dx+\int \{\frac{d}{dx}\times \frac{1}{lnx}$$$$\int dx\}dx+\int \frac{dx}{{\left(lnx\right)}^2}$$$$=xln\left(lnx\right)-\frac{x}{lnx}-\int \frac{dx}{{\left(lnx\right)}^2}+\int \frac{dx}{{\left(lnx\right)}^2}$$$$=xln\left(lnx\right)-\frac{x}{lnx}+C$$

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