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Question Number 13360 by tawa tawa last updated on 19/May/17

$$\int \frac{\mathrm{x}}{\sqrt{3-2\mathrm{x}-{\mathrm{x}}^2}}\mathrm{dx}$$

Answered by ajfour last updated on 19/May/17

$$\mathit{I}=\int \frac{\mathrm{x}+1-1}{\sqrt{4-{(\mathrm{x}+1)}^2}}\mathrm{dx}$$$$-\frac{1}{2}\int \frac{-2(x+1)}{\sqrt{4-{(x+1)}^2}}dx-\int \frac{dx}{\sqrt{4-{(x+1)}^2}}$$$$forfirstintegrallet\mathit{t}=4-{(x+1)}^2$$$$dt=-2(x+1)dx$$$$I=-\frac{1}{2}\int \frac{dt}{\sqrt{t}}-{\sin }^{-1}\left(\frac{x+1}{2}\right)+C_1$$$$I=-\frac{1}{2}\left(2\sqrt{t}\right)-{\sin }^{-1}\left(\frac{x+1}{2}\right)+C$$$$I=-\sqrt{4-{(x+1)}^2}-{\sin }^{-1}\left(\frac{x+1}{2}\right)+C.$$

Commented by tawa tawa last updated on 19/May/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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