Question Number 217225 by hardmath last updated on 06/Mar/25
$$\mathrm{Find}:\:\:\:\:\:\int\:−\:\frac{{d}\mathrm{x}}{\:\sqrt{\mathrm{24x}\:−\:\mathrm{16x}^{\mathrm{2}} \:−\:\mathrm{8}}}\:=\:? \\ $$
Answered by Frix last updated on 06/Mar/25
![−∫(dx/( (√(−16x^2 +24x−8)))) =^([t=sin^(−1) (4x−3)]) =−(1/4)∫dt=−(t/4)=−((sin^(−1) (4x−3))/4)+C](https://www.tinkutara.com/question/Q217226.png)
$$−\int\frac{{dx}}{\:\sqrt{−\mathrm{16}{x}^{\mathrm{2}} +\mathrm{24}{x}−\mathrm{8}}}\:\overset{\left[{t}=\mathrm{sin}^{−\mathrm{1}} \:\left(\mathrm{4}{x}−\mathrm{3}\right)\right]} {=} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{4}}\int{dt}=−\frac{{t}}{\mathrm{4}}=−\frac{\mathrm{sin}^{−\mathrm{1}} \:\left(\mathrm{4}{x}−\mathrm{3}\right)}{\mathrm{4}}+{C} \\ $$