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 54034 by qw last updated on 28/Jan/19

$$\mathrm{If}f\left(x\right)=f(a+b-x),\mathrm{then}\underset{a}{\stackrel{b}{\int }}xf\left(x\right)dx=$$

Commented by maxmathsup by imad last updated on 28/Jan/19

$${\int }_a^{b}xf\left(x\right)dx={\int }_a^{b}xf(a+b-x)dxchangementa+b-x=tgive$$$${\int }_a^{b}xf\left(x\right)dx={\int }_a^b(a+b-t)f\left(t\right)dt={\int }_a^b(a+b)f\left(t\right)-{\int }_a^{b}tf\left(t\right)dt\Rightarrow$$$$2{\int }_a^{b}xf\left(x\right)=(a+b){\int }_a^{b}f\left(x\right)\Rightarrow {\int }_a^{b}xf\left(x\right)dx=\frac{a+b}{2}{\int }_a^{b}f\left(x\right)dx.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com