Question

Integrala aceasta prin parti: $\int\limits^ {} \frac{1}{x} lnx dx$

Answer 1

$\displaystyle \int\dfrac{1}{x}\cdot \ln x\, dx = \int (\ln x)'\cdot \ln x\, dx =\\ \\ = \ln x\cdot \ln x -\int \ln x\cdot (\ln x)'\, dx+C =\\ \\ =\ln^2 x - \int \ln x\cdot \dfrac{1}{x}\, dx+C\\ \\ \\\int\dfrac{1}{x}\cdot \ln x\, dx+ \int \ln x\cdot \dfrac{1}{x}\, dx = \ln^2 x+C\\ \\ \\ 2\int\dfrac{1}{x}\cdot \ln x\, dx = \ln^2 x+C\\ \\ \\ \Rightarrow \int\dfrac{1}{x}\cdot \ln x\, dx = \dfrac{\ln^2 x}{2}+C$