Integrate the following functions- -1-x-x d x

∫1/x √(x)dx =∫1/√(x)(√(x) 1)dx Let √(x) 1=t ⇒1/2√(x)=dt/dx⇒ dx =2 √(x)dt ∴∫1/√(x)(√(x) 1)dx =∫1/√(x)t2√(x)dt =∫2/tdt =2. log |t|+C=2 log |√(x) 1|+C ...

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Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate the...

Question

Integrate the following functions.
$\int \frac{1}{x - \sqrt{x}} d x .$

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Solution

$\int \frac{1}{x - \sqrt{x}} d x = \int \frac{1}{\sqrt{x} (\sqrt{x} - 1)} d x$
Let $\sqrt{x} - 1 = t \Rightarrow \frac{1}{2 \sqrt{x}} = \frac{d t}{d x} \Rightarrow d x = 2 \sqrt{x} d t$
$∴ \int \frac{1}{\sqrt{x} (\sqrt{x} - 1)} d x = \int \frac{1}{\sqrt{x} t} 2 \sqrt{x} d t = \int \frac{2}{t} d t = 2. l o g | t | + C = 2 l o g | \sqrt{x} - 1 | + C$

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