Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 168303 by Mathspace last updated on 07/Apr/22

$$calculate{\int }_0^{1}x\sqrt{1-x^6}dx$$

Answered by peter frank last updated on 07/Apr/22

$$\mathrm{check}\mathrm{Qn}168231$$

Answered by Mathspace last updated on 08/Apr/22

$$thisintegralisresoluble$$$$wedothechangementx=t^{\frac{1}{6}}\Rightarrow$$$${\int }_0^{1}x\sqrt{1-x^6}dx=\frac{1}{6}{\int }_0^1{t}^{\frac{1}{6}}{(1-t)}^{\frac{1}{2}}{t}^{\frac{1}{6}-1}dt$$$$=\frac{1}{6}{\int }_0^1{t}^{\frac{1}{3}-1}{(1-t)}^{\frac{3}{2}-1}dt$$$$weknowB(p,q)={\int }_0^1{t}^{p-1}{(1-t)}^{q-1}dt$$$$=\frac{\mathrm{\Gamma }\left(p\right).\mathrm{\Gamma }\left(q\right)}{\mathrm{\Gamma }(p+q)}\Rightarrow$$$$I=\frac{1}{6}B(\frac{1}{3},\frac{3}{2})=\frac{1}{6}\times \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right).\mathrm{\Gamma }\left(\frac{3}{2}\right)}{\mathrm{\Gamma }(\frac{1}{3}+\frac{3}{2})}$$$$\mathrm{\Gamma }\left(\frac{3}{2}\right)=\mathrm{\Gamma }(\frac{1}{2}+1)=\frac{1}{2}\mathrm{\Gamma }\left(\frac{1}{2}\right)=\frac{\sqrt{\pi }}{2}$$$$\Rightarrow I=\frac{\sqrt{\pi }}{12}.\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{11}{6}\right)}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com