Integration Questions

Question Number 101451 by yahyajan last updated on 02/Jul/20

Commented by Dwaipayan Shikari last updated on 02/Jul/20

$${sin}^{−\mathrm{1}} {x}\int\mathrm{1}{dx}−\int\frac{{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\$$$${xsin}^{−\mathrm{1}} {x}\:+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{−\mathrm{2}{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}\:\:\:={xsin}^{−\mathrm{1}} {x}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+{c} \\$$

Answered by floor(10²Eta[1]) last updated on 02/Jul/20

$$\int{f}^{−\mathrm{1}} \left({x}\right){dx}={xf}^{−\mathrm{1}} \left({x}\right)−{F}\circ{f}^{−\mathrm{1}} \left({x}\right)+{C} \\$$$${let}\:\:{f}^{−\mathrm{1}} \left({x}\right)={sin}^{−\mathrm{1}} \left({x}\right)\Rightarrow{f}\left({x}\right)={sin}\left({x}\right) \\$$$$\Rightarrow{F}\left({x}\right)=\int{sin}\left({x}\right){dx}=−{cos}\left({x}\right) \\$$$$\int{sin}^{−\mathrm{1}} \left({x}\right){dx}={xsin}^{−\mathrm{1}} \left({x}\right)+{cos}\left({sin}^{−\mathrm{1}} \left({x}\right)\right)+{C} \\$$$${cos}\left({sin}^{−\mathrm{1}} \left({x}\right)\right)={cos}\left(\alpha\right)\:\left[{sin}^{−\mathrm{1}} \left({x}\right)=\alpha\Rightarrow{sin}\left(\alpha\right)={x}\right] \\$$$${drawing}\:{a}\:{right}\:{triangle}\:{with}\:{angle}\:\alpha: \\$$$${sin}\left(\alpha\right)={x}\Rightarrow\frac{{opp}}{{hyp}}=\frac{{x}}{\mathrm{1}}\Rightarrow{opp}={x}\:{and}\:{hyp}=\mathrm{1} \\$$$$\Rightarrow{adj}=\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\Rightarrow{cos}\left(\alpha\right)=\sqrt{\mathrm{1}−{x}^{\mathrm{2}} } \\$$$$\int{sin}^{−\mathrm{1}} \left({x}\right){dx}={xsin}^{−\mathrm{1}} \left({x}\right)+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+{C} \\$$

Answered by  M±th+et+s last updated on 02/Jul/20

$${I}=\int{sin}^{−\mathrm{1}} \left({x}\right){dx} \\$$$${I}=\int{sin}^{−\mathrm{1}} \left({x}\right)+\frac{{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}−\int\frac{{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\$$$${I}=\int{d}\left({xsin}^{−\mathrm{1}} \left({x}\right)\right)−\int\frac{{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\$$$${I}={xsin}^{−\mathrm{1}} \left({x}\right)+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+{c} \\$$