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Question:
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Integrate the function
$\frac{1}{x-\sqrt{x}}$
$\frac{1}{x-\sqrt{x}}$
Solution:
1734 Upvotes
Verified Answer
$\int \frac{1}{x-\sqrt{x}} d x=\int \frac{1}{\sqrt{x}(\sqrt{x}-1)} d x=\mathrm{I}$
Let $\sqrt{x}-1=t \Rightarrow \frac{1}{2} x^{-\frac{1}{2}} d x=d t$
$\mathrm{I}=2 \int \frac{d t}{t}=2 \log t+\mathrm{C}=2 \log
(\sqrt{x}-1)+\mathrm{C}$
Let $\sqrt{x}-1=t \Rightarrow \frac{1}{2} x^{-\frac{1}{2}} d x=d t$
$\mathrm{I}=2 \int \frac{d t}{t}=2 \log t+\mathrm{C}=2 \log
(\sqrt{x}-1)+\mathrm{C}$
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