Integrate the functions in Exercises: 1 x- x - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{1}{\text{x}-\sqrt{\text{x}}}$

✓

Answer

$\text{Let I}=\int\frac{1}{\text{x }-\sqrt{\text{x}}}\text{ dx} \ \ \ \ \ \ \ \dots\text{(i)}$
Putting $\sqrt{ \text{x}}=\text{t}\ \ \ \ \Rightarrow \ \ \ \ \text{x}=\text{t}^2\ \ \ \Rightarrow \ \ \ \frac{\text{dx}}{\text{dt}}=2\text{t}\ \ \ \ \Rightarrow \ \ \ \text{dx}=2\text{t}\text{ dt} $
$\therefore \ \ \ \ $From eq. (i),$\text{ I}=\int\frac{1}{\text{t}^2-\text{t}}\text{2t}\text{ dt} =2\int{\frac{\text{t}}{\text{t(t - 1)}}}\text{ dt}$
$=2\int\frac{1}{(\text{t }-1)}\text{ dt}=2\log\mid\text{t}-1\mid+\text{ c}$
$=2\log\begin{vmatrix}\sqrt{\text{x}} -1\end{vmatrix}+\text{c} $

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