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Integrate the function
$\frac{x+2}{\sqrt{4 x-x^2}}$
$\frac{x+2}{\sqrt{4 x-x^2}}$
Solution:
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Verified Answer
$\mathrm{I}=\int \frac{x-2}{\sqrt{4-(x-2)^2}} d x+4 \int \frac{d x}{\sqrt{4-(x-2)^2}}$
$=\mathrm{I}_1+4 \sin ^{-1} \frac{x-2}{2}+\mathrm{C}$
For $\mathrm{I}_1, \operatorname{put}(x-2)^2=\mathrm{t} \Rightarrow 2(x-2) d x=d t$
$\therefore \quad \mathrm{I}_1=\frac{1}{2} \int \frac{d t}{\sqrt{4-t}}=\sqrt{4-t}$
$\therefore \quad \mathrm{I}=\sqrt{4-(x-2)^2}+4 \sin ^{-1} \frac{x-2}{2}+C$
$=\mathrm{I}_1+4 \sin ^{-1} \frac{x-2}{2}+\mathrm{C}$
For $\mathrm{I}_1, \operatorname{put}(x-2)^2=\mathrm{t} \Rightarrow 2(x-2) d x=d t$
$\therefore \quad \mathrm{I}_1=\frac{1}{2} \int \frac{d t}{\sqrt{4-t}}=\sqrt{4-t}$
$\therefore \quad \mathrm{I}=\sqrt{4-(x-2)^2}+4 \sin ^{-1} \frac{x-2}{2}+C$
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