Integrate the functions in Exercises: (1+ )^2 x - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{(1+\log\text{x})^2}{\text{x}}$

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Answer

$\text{Let I}=\int\frac{(1+\log\text{x})^2}{\text{x}}\text{ dx} \ \ \ \ ... \text{(i)}$
Putting $1+\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), $\text{I}=\int\text{t}^2\text{ dt}=\frac{\text{t}^3}{3}+\text{c}= \frac{1}{3}(1+\log\text{x})^3+\text{c}$

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