Evaluate: (i) `int1/(sqrt(1-cos2x))\ dx` (ii) `int1/(sqrt(1+cosx))\ dx`

Explanation

(i)
`cos 2 x=cos ^{2} x-sin ^{2} x`

`cos ^{2} x+sin ^{2} x=1`

`Rightarrow int frac{1}{sqrt{sin ^{2} x+cos ^{2} x-(-sin ^{2} x+cos ^{2} x)}} d x `

`Rightarrow int frac{1}{sqrt{sin ^{2} x+cos ^{2} x+sin ^{2} x-cos ^{2} x}} d x `

`Rightarrow int frac{1}{sqrt{2 sin ^{2} x}} d x `

`Rightarrow int frac{1}{sqrt{2} sin x} d x `

`frac{1}{sqrt{2}} int cosec x d x `

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