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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate :$\int \sqrt {4x-x^2}dx$

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

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A)
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$\int \sqrt {-(4-4x+x^2)+4}.dx$
$\int \sqrt {(2)^2 -(x-2)^2 }.dx$
$(x-2)=t=> dx=dt$
=>$\int \sqrt {(2)^2 -(t)^2 }.dt$
=> $\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.
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