Integration Questions

Question Number 170578 by mnjuly1970 last updated on 27/May/22

$$\\$$ $$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{{x}}\:{sin}^{\:−\mathrm{1}} \left(\:{x}\:\right){dx}\:=\:? \\$$ $$\\$$

Answered by Mathspace last updated on 27/May/22

$${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{x}}{arcsin}\left({x}\right){dx} \\$$ $${x}={sint}\:\Rightarrow{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sint}}{t}\:\:{cost}\:{dt} \\$$ $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {t}\left({cost}\sqrt{{sint}}\right){dt} \\$$ $${let}={u}={t}\:{and}\:{v}^{'} ={cost}\:\sqrt{{sint}} \\$$ $$\Rightarrow{v}=\frac{\mathrm{2}}{\mathrm{3}}\left({sint}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \:\Rightarrow \\$$ $${I}=\left[\frac{\mathrm{2}}{\mathrm{3}}{t}\left({sint}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −\frac{\mathrm{2}}{\mathrm{3}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sint}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dt} \\$$ $$=\frac{\pi}{\mathrm{3}}−\frac{\mathrm{2}}{\mathrm{3}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sint}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dt} \\$$ $${but}\:{we}\:{know}\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{\mathrm{2}{p}−\mathrm{1}} {t}\:{sin}^{\mathrm{2}{q}−\mathrm{1}} {tdt} \\$$ $$={B}\left({p},{q}\right)=\frac{\Gamma\left({p}\right).\Gamma\left({q}\right)}{\Gamma\left({p}+{q}\right)}\:\Rightarrow \\$$ $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sint}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dt}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sint}\right)^{\mathrm{2}.\frac{\mathrm{5}}{\mathrm{4}}−\mathrm{1}} \:{cos}^{\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{1}} {dt} \\$$ $$=\frac{\mathrm{1}}{\mathrm{2}}{B}\left(\frac{\mathrm{5}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{2}}\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\$$ $$=\frac{\sqrt{\pi}}{\mathrm{2}}×\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{7}}{\mathrm{4}}\right)}\:\Rightarrow \\$$ $${I}=\frac{\pi}{\mathrm{3}}−\frac{\sqrt{\pi}}{\mathrm{3}}×\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{7}}{\mathrm{4}}\right)} \\$$

Commented bymnjuly1970 last updated on 28/May/22

$${verynice}\:\:{solution}\:\:{thank}\:{you}.. \\$$

Commented byTawa11 last updated on 08/Oct/22

$$\mathrm{Great}\:\mathrm{sir} \\$$