Question

íntegra ln x ao quadrado dividido por x dx

Answer 1

Pelas propriedades de logaritmo, $log(x^{2})=2\cdot log(x)$
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$\displaystyle\int\dfrac{log(x^{2})}{x}dx=\int2\dfrac{log(x)}{x}dx\\\\\int\dfrac{log(x^{2})}{x}dx=2\int \dfrac{log(x)}{x}dx$

Fazendo $u=log(x)~\rightarrow~du=\frac{1}{x}dx$:

$\displaystyle\int\dfrac{log(x^{2})}{x}dx=2\int log(x)\dfrac{1}{x}dx\\\\\\\int\dfrac{log(x^{2})}{x}dx=2\int udu\\\\\\\int\dfrac{log(x^{2})}{x}dx=2\dfrac{u^{2}}{2}+C\\\\\\\int\dfrac{log(x^{2})}{x}dx=u^{2}+C\\\\\\\boxed{\boxed{\int\dfrac{log(x^{2})}{x}dx=(log~x)^{2}+C}}$