Evaluate the following intregals: 1 x (2+ ) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}\text{ dx}$

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Answer

Let $\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}=\frac{\text{A}}{\text{x}\log\text{x}}+\frac{\text{B}}{\text{x}(2+\log\text{x})}$
$\Rightarrow1=\text{A}(2+\log\text{x})+\text{B}\log\text{x}$
Put $x = 1$
$\Rightarrow1=2\text{A}\Rightarrow\text{A}=\frac{1}{2}$
Put $x = 10^{-2}$
$\Rightarrow1=-2\text{B}\Rightarrow\text{B}=-\frac{1}{2}$
Thus,
$\text{I}=\frac{1}{2}\int\frac{\text{dx}}{\text{x}\log\text{x}}+\Big(-\frac{1}{2}\Big)\int\frac{\text{dx}}{\text{x}(2+\log\text{x})}$
$=\frac{1}{2}\log|\log\text{x}|-\frac{1}{2}\log|2+\log\text{x}|+\text{C}$
$\text{I}=\frac{1}{2}\log\Big|\frac{\log\text{x}}{2+\log\text{x}}\Big|+\text{C}$

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