Question
Evaluate the following integrals:
$\int\frac{\log\text{x}^2}{\text{x}}\text{dx}$
$\int\frac{\log\text{x}^2}{\text{x}}\text{dx}$
✓
Answer
$\int\frac{\log\text{x}^2\text{dx}}{\text{x}}$
$=\int\frac{2\log\text{x}}{\text{x}}\text{dx}$
$=2\int\frac{\log\text{x}}{\text{x}}\text{dx}$
$\text{Let }\log\text{x}=\text{t}$
$\Rightarrow\frac{1}{\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{1}{\text{x}}\text{dx}=\text{dt}$
$\text{Now, }2\int\frac{\log\text{x}}{\text{x}}\text{dx}$
$=2\int\text{t dt}$
$2\Big[\frac{\text{t}^2}{2}\Big]+\text{C}$
$=\text{t}^2+\text{C}$
$=(\log\text{x})^2+\text{C}$
$=\int\frac{2\log\text{x}}{\text{x}}\text{dx}$
$=2\int\frac{\log\text{x}}{\text{x}}\text{dx}$
$\text{Let }\log\text{x}=\text{t}$
$\Rightarrow\frac{1}{\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{1}{\text{x}}\text{dx}=\text{dt}$
$\text{Now, }2\int\frac{\log\text{x}}{\text{x}}\text{dx}$
$=2\int\text{t dt}$
$2\Big[\frac{\text{t}^2}{2}\Big]+\text{C}$
$=\text{t}^2+\text{C}$
$=(\log\text{x})^2+\text{C}$
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