Question
Evalute the following integrals:
$\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$
$\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$
Let
$\text{I}=\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$Putting
$\log\tan\text{x}=\text{t}$$\Rightarrow\frac{1}{\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{\text{dx}}{\text{x}}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{3+\text{t}}$
$=\log|3+\text{t}|+\text{C}$
$=\log|3+\log\text{x}|+\text{C}\ \big[\because\text{t}=\log\text{x}\big]$
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