Integrate :$\int \sqrt {4x-x^2}dx$

Question

\[\begin {array} {1 1} (a)\;\frac{(x-2)}{2}\sqrt {4x-2x^2}+2 \sin ^{-1} \bigg(\frac{x-2}{2}\bigg)+c \\ (b)\;\frac{(x-2)}{2}\sqrt {4x-x^2}+2 \sin ^{-1} \bigg(\frac{x-2}{2}\bigg)+c \\ (c)\;\frac{-(x-2)}{2}\sqrt {4x-x^2}+2 \sin ^{-1} \bigg(\frac{x-2}{2}\bigg)+c \\ (d)\;None \end {array}\]

Answer 1

$\int \sqrt {-(4-4x+x^2)+4}.dx$

$\int \sqrt {(2)^2 -(x-2)^2 }.dx$

$(x-2)=t=> dx=dt$

=>$\int \sqrt {(2)^2 -(t)^2 }.dt$

=> $\large\frac{t}{2} . $$\sqrt {(2)^2-t^2}+\large\frac{4}{2}.$$ \sin ^{-1} \bigg(\large\frac{t}{2}\bigg)$$+c$

$\large\frac{(x-2)}{2}$$\sqrt {4x-x^2}+2 \sin ^{-1} \bigg(\frac{x-2}{2}\bigg)+c $

Hence b is the correct answer.