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 167922 by peter frank last updated on 29/Mar/22

$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{Q}\mathrm{if}$$$${\int }_0^{\mathrm{Q}}\sqrt{\mathrm{cosec}\theta -1}\mathrm{d}\theta =\ln \left(\frac{3+2\sqrt{2}}{2}\right)$$$$$$

Commented by MJS_new last updated on 29/Mar/22

$$dx?$$

Commented by peter frank last updated on 29/Mar/22

$$\mathrm{d}\theta$$

Answered by MJS_new last updated on 29/Mar/22

$$\int \sqrt{\csc \theta -1}d\theta =$$$$[t=\sqrt{\csc \theta -1}\to d\theta =-\frac{2dt}{(t^2+1)\sqrt{t^2+2}}]$$$$=-2\int \frac{t}{(t^2+1)\sqrt{t^2+2}}=$$$$[u=\frac{t+\sqrt{t^2+2}}{\sqrt{2}}\to dt=\frac{\sqrt{t^2+2}}{u}]$$$$=-2\sqrt{2}\int \frac{u^2-1}{u^4+1}du=$$$$=\int \frac{2u+\sqrt{2}}{u^2+\sqrt{2}u+1}-\frac{2u-\sqrt{2}}{u^2-\sqrt{2}u+1})du=$$$$=\ln \frac{u^2+\sqrt{2}u+1}{u^2-\sqrt{2}u+1}=$$$$...$$$$=\ln (1+2\sin x+2\sqrt{(1+\sin x)\sin x})+C$$$$\mathrm{this}\mathrm{equals}0\mathrm{with}x=0$$$$\Rightarrow$$$$1+2\sin Q+2\sqrt{(1+\sin Q)\sin Q}=\frac{3+2\sqrt{2}}{2}$$$$\Rightarrow \sin Q=\frac{11-6\sqrt{2}}{8}$$$$...$$

Answered by peter frank last updated on 29/Mar/22

$${\int }_0^{\mathrm{Q}}\sqrt{\mathrm{cosec}\theta -1}\mathrm{d}\theta ={\int }_0^{\mathrm{Q}}\sqrt{\frac{1-\sin \theta }{\sin \theta }}\mathrm{d}\theta$$$${\int }_0^{\mathrm{Q}}\sqrt{\frac{1-\sin \theta }{\sin \theta }}\mathrm{d}\theta =\ln \left(\frac{3+2\sqrt{2}}{2}\right)$$$$$$$$.......$$

Answered by peter frank last updated on 30/Mar/22

Answered by peter frank last updated on 30/Mar/22

Commented by peter frank last updated on 30/Mar/22

$$\mathrm{thank}\mathrm{you}\mathrm{Mjs}\mathrm{for}\mathrm{you}\mathrm{idea}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com