find-dx-4x-x-2-

Question Number 33329 by prof Abdo imad last updated on 14/Apr/18

f i n d \int \frac{d x}{\sqrt{4 x - x^{2}}} .

Commented by prof Abdo imad last updated on 19/Apr/18

I = \int \frac{d x}{\sqrt{- x^{2} + 4 x}} = \int \frac{d x}{\sqrt{- (x^{2} - 4 x + 4) + 4}}

= \int \frac{d x}{\sqrt{4 - {(x - 2)}^{2}}} . c h a n g e m e n t x - 2 = 2 s i n θ

= \int \frac{2 c o s θ d θ}{\sqrt{4 - 4 {s i n}^{2} θ}} = \int d θ = θ + c

= a r c s i n (\frac{x}{2} - 1) + c .