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Integration of Irrational Algebraic Fractions - 1

∫√cot x - √ta...

Question

$\int \frac{\sqrt{c o t x} - \sqrt{t a n x}}{\sqrt{2} (c o s x + s i n x)} d x$ equals to

A

s e c^{- 1} (s i n x + c o s x) + c

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B

s e c^{- 1} (s i n x - c o s x) + c

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C

l n | (s i n x + c o s x) + \sqrt{s i n 2 x} | + c

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D

l n | (s i n x - c o s x) + \sqrt{s i n 2 x} | + c

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Solution

The correct option is C $l n | (s i n x + c o s x) + \sqrt{s i n 2 x} | + c$
Let $I = \int \frac{\sqrt{cot x} - \sqrt{tan x}}{\sqrt{2} (c o s x + s i n x)} d x$
$= \int \frac{(c o s x - s i n x)}{\sqrt{2} \sqrt{c o s x s i n x} (c o s x + s i n x)} d x$
$= \int \frac{(c o s x - s i n x)}{\sqrt{s i n 2 x} (c o s x + s i n x)} d x$
Put $c o s x + s i n x = t \Rightarrow (- s i n x + c o s x) d x = d t$

And ${(c o s x + s i n x)}^{2} = t^{2} \Rightarrow 2 s i n x c o s x = t^{2} - 1 \Rightarrow s i n 2 x = t^{2} - 1$

$∴ I = \int \frac{d t}{\sqrt{t^{2} - 1}}$

$= log (\sqrt{t^{2} - 1} + t) + c$

$= l n | (s i n x + c o s x) + \sqrt{s i n 2 x} | + c$

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