Mathematics

Evaluate the following integral:
$$\int { \cfrac { \log { { x }^{ 2 } }  }{ x }  } dx\quad $$


SOLUTION
$$\displaystyle\int{\dfrac{\log{{x}^{2}}}{x}dx}$$

$$=\displaystyle\int{\dfrac{2\log{x}}{x}dx}$$

$$=2\displaystyle\int{\dfrac{\log{x}}{x}dx}$$

Let $$t=\log{x}\Rightarrow\,dt=\dfrac{dx}{x}$$

$$=2\displaystyle\int{t\,dt}$$

$$=2\times\dfrac{{t}^{2}}{2}+c$$

$$={t}^{2}+c$$

$$={\left(\log{x}\right)}^{2}+c$$ where $$t=\log{x}$$
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Subjective Medium Published on 17th 09, 2020
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