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Question Number 190318 by Rupesh123 last updated on 31/Mar/23

Answered by som(math1967) last updated on 31/Mar/23

$$let2x-x^2-1=t^2$$$$(2-2x)dx=2tdt$$$$\int \frac{2tdt}{t}=2t+C=2\sqrt{2x-x^2-1}+C$$

Answered by hknkrc46 last updated on 01/Apr/23

$$▸2\mathit{x}-{\mathit{x}}^2-1=-{(\mathit{x}-1)}^2$$$$\mathrm{♡}\sqrt{2\mathit{x}-{\mathit{x}}^2-1}=\sqrt{-{(\mathit{x}-1)}^2}$$$$=\sqrt{-1}\cdot \sqrt{{(\mathit{x}-1)}^2}=\mathit{i}\mid \mathit{x}-1\mid$$$$=\mp \mathit{i}(1-\mathit{x})$$$$▸\int \frac{2-2\mathit{x}}{\sqrt{2\mathit{x}-{\mathit{x}}^2-1}}\mathit{d}\mathit{x}$$$$=\int \frac{2(1-\mathit{x})}{\mp \mathit{i}(1-\mathit{x})}\mathit{d}\mathit{x}=\mp 2\mathit{i}\int \mathit{d}\mathit{x}$$$$=\mathit{c}\mp 2\mathit{i}\mathit{x}$$

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