dx-cos-x-cosec-x-

Question Number 152291 by iloveisrael last updated on 27/Aug/21

\int \frac{d x}{\cos x + cosec x} = ?

Answered by puissant last updated on 27/Aug/21

I = \int \frac{d x}{c o s x + c o s e c x}

= \int \frac{s i n x}{c o s x s i n x + 1} d x = \int \frac{2 s i n x}{s i n 2 x + 2} d x

= \int \frac{s i n x + c o s x}{s i n 2 x + 2} d x + \int \frac{s i n x - c o s x}{s i n 2 x + 2} d x

t = s i n x - c o s x \to d t = (s i n x + c o s x) d x

t^{2} = 1 - s i n 2 x \Rightarrow s i n 2 x = 1 - t^{2}

u = s i n x + c o s x \to - d u = (s i n x - c o s x) d x

u^{2} = 1 + s i n 2 x \to s i n 2 x = u^{2} - 1

\Rightarrow I = \int \frac{d t}{1 - t^{2} + 2} - \int \frac{d u}{u^{2} - 1 + 2}

= \int \frac{d t}{{(\sqrt{3})}^{2} - {(t)}^{2}} - \int \frac{d u}{{(u)}^{2} + {(1)}^{2}}

= \frac{1}{2 \sqrt{3}} l n ∣ \frac{\sqrt{3} - t}{\sqrt{3} + t} ∣ - a r c t a n (u) + C

∴∵ I = \frac{1}{2 \sqrt{3}} l n ∣ \frac{\sqrt{3} - (s i n x - c o s x)}{\sqrt{3} + (s i n x - c o s x)} ∣ - a r c t a n (s i n x + c o s x) + C . .

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