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Question Number 218163 by hardmath last updated on 31/Mar/25

$$\mathrm{Find}:\int \frac{\mathrm{x}}{\sqrt{48-2\mathrm{x}-{\mathrm{x}}^2}}\mathrm{dx}=?$$

Answered by efronzo1 last updated on 31/Mar/25

$$\mathrm{I}=\int \frac{\mathrm{x}}{\sqrt{49-{(\mathrm{x}+1)}^2}}\mathrm{dx}$$$$\mathrm{x}+1=7\sin \mathrm{u}$$$$\mathrm{dx}=7\cos \mathrm{u}\mathrm{du}$$$$\mathrm{I}=\int \frac{7\sin \mathrm{u}-1}{7\sqrt{1-\sin {}^2\mathrm{u}}}.7\cos \mathrm{u}\mathrm{du}$$$$=\int (7\sin \mathrm{u}-1)\mathrm{du}$$$$=-7\cos \mathrm{u}-\mathrm{u}+\mathrm{c}$$$$=-7\left(\frac{\sqrt{48-2\mathrm{x}-{\mathrm{x}}^2}}{7}\right)-\arcsin \left(\frac{\mathrm{x}+1}{7}\right)+\mathrm{c}$$$$=-\sqrt{48-2\mathrm{x}-{\mathrm{x}}^2}-\arcsin \left(\frac{\mathrm{x}+1}{7}\right)+\mathrm{c}$$

Commented by hardmath last updated on 31/Mar/25

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{ser}$$

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