Evaluate: (i) `intsqrt((1-sin2x)/(1+sin2x))\ dx` (ii) `int(e^(3x))/(e^(3x)+1)\ dx`

Explanation

(i)
`int sqrt{frac{1-sin 2 x}{1+sin 2 x}} d x `

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

`=sqrt{frac{(cos x-sin x)^{2}}{(cos x+sin x)^{2}}} d x `

`=int frac{cos x-sin x}{cos x+sin x} d x `

`=int frac{1-tan x}{1+tan x} d x `

`=int tan (frac{pi}{4}-x) d x `

`=frac{1}{-1} ln |sec (frac{pi}{4}-x)|[because int tan (a x+b) d x=frac{1}{a} ln |sec (a x+b)|+C] `

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