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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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Consider the integrals $I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$ and $I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$