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 130448 by physicstutes last updated on 25/Jan/21

Given that f(x) = f(π−x), prove that ∫_0 ^π xf(x)dx = (π/2)∫_0 ^π f(x)dx. please what are different methods to approach this question?

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{f}\left(\pi−{x}\right),\:\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx}\:=\:\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}. \\ $$$$\mathrm{please}\:\mathrm{what}\:\mathrm{are}\:\mathrm{different}\:\mathrm{methods}\:\mathrm{to}\:\mathrm{approach}\:\mathrm{this}\:\mathrm{question}? \\ $$

Answered by TheSupreme last updated on 25/Jan/21

∫_0 ^π xf(x)dx u=π−x du=−dx ∫_π ^0 (π−u)f(π−u)(−du)=∫_0 ^π xf(x) ∫_0 ^π πf(π−u)du−∫_0 ^π uf(π−u)du=∫_0 ^π xf(x)dx than, f(u)=f(π−u); set I=∫_0 ^π xf(x)dx ∫_0 ^π πf(u)du=2∫_0 ^π uf(u)du ∫_0 ^π (π/2)f(u)du=∫_0 ^π uf(u)du

$$\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx} \\ $$$${u}=\pi−{x} \\ $$$${du}=−{dx} \\ $$$$\int_{\pi} ^{\mathrm{0}} \left(\pi−{u}\right){f}\left(\pi−{u}\right)\left(−{du}\right)=\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right) \\ $$$$\int_{\mathrm{0}} ^{\pi} \pi{f}\left(\pi−{u}\right){du}−\int_{\mathrm{0}} ^{\pi} {uf}\left(\pi−{u}\right){du}=\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx} \\ $$$${than},\:{f}\left({u}\right)={f}\left(\pi−{u}\right);\:{set}\:{I}=\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx} \\ $$$$\int_{\mathrm{0}} ^{\pi} \pi{f}\left({u}\right){du}=\mathrm{2}\int_{\mathrm{0}} ^{\pi} {uf}\left({u}\right){du} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\pi}{\mathrm{2}}{f}\left({u}\right){du}=\int_{\mathrm{0}} ^{\pi} {uf}\left({u}\right){du} \\ $$$$ \\ $$

Commented by physicstutes last updated on 25/Jan/21

thanks sir

$$\mathrm{thanks}\:\mathrm{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com