Integrate the functions in Exercises: 1 x( )^m ,x>0 - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{1}{\text{x}(\log\text{x})^\text{m}},\text{x}>0$

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Answer

$\text{Let I}=\int\frac{1}{\text {x}(\log \text{x})^\text{m}} \text{ dx}=\int\frac{\frac{1}{\text{x}}\text{dx}}{(\log \text{x)}^\text{m}} \ \ \ \ ....\text{(i)} $
Putting $\log\text{x}=\text{t}\ \ \ \Rightarrow \ \ \ \ \frac {1}{\text{x}}=\frac{\text{dt}}{\text {dx}}\ \ \ \ \Rightarrow \ \ \ \ \frac{\text{dx}}{\text{x}}=\text{dt}$
$\therefore \ \ \ \ \ $ From eq. (i), $\text{I}=\int\frac{\text{dt}}{\text{t}^{\text{m}}}=\int\text{t}^\text{-m}\text{ dt}=\frac{\text{t}^{-\text{m}+1}}{-\text {m + 1}}+\text{c}$
$=\frac{(\log\text{x})^{1-\text {m}}}{1-\text{m}}+\text{c} $

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