find-x-1-3-x-dx-

Question Number 45231 by maxmathsup by imad last updated on 10/Oct/18

f i n d \int \sqrt{(x - 1) (3 - x)} d x

Commented by maxmathsup by imad last updated on 12/Oct/18

l e t A = \int \sqrt{(x - 1) (3 - x)} d x \Rightarrow A = \int \sqrt{3 x - x^{2} - 3 + x} d x = \int \sqrt{- x^{2} + 4 x - 3} d x

= \int \sqrt{- (x^{2} - 4 x + 4 - 4) - 3} d x = \int \sqrt{1 - {(x - 2)}^{2}} d x c h a n g e m e n t

x - 2 = s i n t g i v e A = \int c o s t c o s t d t = \int \frac{1 + c o s (2 t)}{2} d t

= \frac{t}{2} + \frac{1}{4} s i n (2 t) + c = \frac{t}{2} + \frac{1}{2} s i n (t) c o s (t) + c

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

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Oct/18

\int \sqrt{3 x - x^{2} - 3 + x d x}

\int \sqrt{- 3 + 4 x - x^{2}} d x

\int \sqrt{- 3 + 4 - 4 + 4 x - x^{2}} d x

\int \sqrt{1 - {(x - 2)}^{2}} d x

u s e f o r m u l a \int \sqrt{a^{2} - x^{2}} d x

\frac{x - 2}{2} \sqrt{1 - {(x - 2)}^{2}} + \frac{1}{2} {s i n}^{- 1} (\frac{x - 2}{1}) + c

Answered by ajfour last updated on 11/Oct/18

l e t x - 2 = t

\int \sqrt{(x - 1) (3 - x)} d x = \int \sqrt{(t + 1) (1 - t)} d t

= \int \sqrt{1 - t^{2}} d t

= \frac{t}{2} \sqrt{1 - t^{2}} + \frac{1}{2} \sin^{- 1} t + c

= (\frac{x - 2}{2}) \sqrt{(x - 1) (3 - x)} + \frac{1}{2} \sin^{- 1} (x - 2) + c .

Commented by Necxx last updated on 12/Oct/18

h m m m \dots . I c a u g h t i t . T h a n k s

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