Question and Answers Forum

All Questions      Topic List

UNKNOWN Questions

Previous in All Question      Next in All Question      

Previous in UNKNOWN      Next in UNKNOWN      

Question Number 59976 by soufiane last updated on 16/May/19

$$\underset{0}{\stackrel{\pi /2}{\int }}\frac{f\left(x\right)}{f\left(x\right)+f(\frac{\pi }{2}-x)}dx=$$

Answered by tanmay last updated on 16/May/19

$$I={\int }_0^{\frac{\pi }{2}}\frac{f\left(x\right)}{f\left(x\right)+f(\frac{\pi }{2}-x)}dx$$$$t=\frac{\pi }{2}-x$$$$dt=-dx$$$$I={\int }_{\frac{\pi }{2}}^0\frac{f(\frac{\pi }{2}-t)}{f(\frac{\pi }{2}-t)+f\left(t\right)}\times -dt$$$$I={\int }_0^{\frac{\pi }{2}}\frac{f(\frac{\pi }{2}-t)}{f(\frac{\pi }{2}-t)+f\left(t\right)}dt\to {\int }_0^{\frac{\pi }{2}}\frac{f(\frac{\pi }{2}-x)}{f(\frac{\pi }{2}-x)+f\left(x\right)}dx$$$$now{\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f\left(t\right)dt$$$$so$$$$2I={\int }_0^{\frac{\pi }{2}}\frac{f\left(x\right)}{f\left(x\right)+f(\frac{\pi }{2}-x)}dx+{\int }_0^{\frac{\pi }{2}}\frac{f(\frac{\pi }{2}-x)}{f(\frac{\pi }{2}-x)+f\left(x\right)}dx$$$$2I={\int }_0^{\frac{\pi }{2}}dx$$$$2I=\frac{\pi }{2}\to I=\frac{\pi }{4}$$$$◼directformula$$$${\int }_a^{b}f\left(x\right)dx={\int }_0^{b}f(a+b-x)dx$$$$I={\int }_0^{\frac{\pi }{2}}\frac{f\left(x\right)}{f\left(x\right)+f(\frac{\pi }{2}-x)}dx$$$$I={\int }_0^{\frac{\pi }{2}}\frac{f(\frac{\pi }{2}-x)}{f(\frac{\pi }{2}-x)+f\left(x\right)}dx$$$$2I={\int }_0^{\frac{\pi }{2}}\frac{f(\frac{\pi }{2}-x)+f\left(x\right)}{f\left(x\right)+f(\frac{\pi }{2}-x)}dx$$$$2I={\int }_0^{\frac{\pi }{2}}dx$$$$I=\frac{\pi }{4}$$

Answered by Prithwish sen last updated on 16/May/19

$$\mathrm{I}={\int }_0^{\frac{\pi }{2}}\frac{\mathrm{f}(\frac{\pi }{2}-\mathrm{x})}{\mathrm{f}(\frac{\pi }{2}-\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx}$$$$\therefore 2\mathrm{I}={\int }_0^{\frac{\pi }{2}}\frac{\mathrm{f}(\frac{\pi }{2}-\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}(\frac{\pi }{2}-\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx}$$$$={\int }_0^{\frac{\pi }{2}}\mathrm{dx}=\frac{\pi }{2}$$$$\therefore \mathrm{I}=\frac{\pi }{4}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com