1-find-dx-x-1-x-2-2-calculate-0-1-dx-x-1-x-2-

Question Number 64419 by turbo msup by abdo last updated on 17/Jul/19

1) f i n d \int \frac{d x}{x - \sqrt{1 - x^{2}}}

2) c a l c u l a t e \int_{0}^{1} \frac{d x}{x - \sqrt{1 - x^{2}}}

Commented by mathmax by abdo last updated on 18/Jul/19

1) l e t A = \int \frac{d x}{x - \sqrt{1 - x^{2}}} c h a n g e m e n t x = s i n θ g i v e

A = \int \frac{c o s θ}{s i n θ - c o s θ} d θ = \int \frac{c o s θ}{c o s θ (t a n θ - 1)} d θ

= \int \frac{d θ}{t a n θ - 1} =_{t a n θ = t} \int \frac{d t}{(1 + t^{2}) (t - 1)} l e t d e c o m p o s e

F (t) = \frac{1}{(t - 1) (t^{2} + 1)} \Rightarrow F (t) = \frac{a}{t - 1} + \frac{b t + c}{t^{2} + 1}

a = {l i m}_{t \to 1} (t - 1) F (t) = \frac{1}{2}

{l i m}_{t \to + \infty} t F (t) = 0 = a + b \Rightarrow b = - \frac{1}{2} \Rightarrow F (t) = \frac{1}{2 (t - 1)} + \frac{- \frac{1}{2} t + c}{t^{2} + 1}

F (0) = - 1 = - \frac{1}{2} + c \Rightarrow c = - \frac{1}{2} \Rightarrow F (t) = \frac{1}{2 (t - 1)} - \frac{1}{2} \frac{t + 1}{t^{2} + 1} \Rightarrow

A = \frac{1}{2} \int \frac{d t}{t - 1} - \frac{1}{2} \int \frac{t + 1}{t^{2} + 1} d t + c

= \frac{1}{2} l n ∣ t - 1 ∣ - \frac{1}{4} l n (t^{2} + 1) - \frac{1}{2} a r c t a n (t) + c

w e h a v e t = t a n θ a n d θ = a r c s i n x \Rightarrow

A = \frac{1}{2} l n ∣ t a n (a r c s i n x) - 1 ∣ - \frac{1}{4} l n (1 + {t a n}^{2} (a r c s i n x)) - \frac{1}{2} a r c s i n x + c