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 87881 by Rio Michael last updated on 06/Apr/20

 ∫_(−∞) ^( +∞) (1/x) dx =

$$\:\int_{−\infty} ^{\:+\infty} \frac{\mathrm{1}}{{x}}\:{dx}\:=\: \\ $$

Commented by mathmax by abdo last updated on 06/Apr/20

the function x→(1/x) is odd ⇒∫_(−∞) ^(+∞)  (dx/x)=0   also we can write  ∫_(−∞) ^(+∞)  (dx/x) =lim_(ξ→+∞)   ∫_(−ξ) ^ξ  (dx/x)  =lim_(ξ→+∞)    [ln∣x∣]_(−ξ) ^ξ  =0

$${the}\:{function}\:{x}\rightarrow\frac{\mathrm{1}}{{x}}\:{is}\:{odd}\:\Rightarrow\int_{−\infty} ^{+\infty} \:\frac{{dx}}{{x}}=\mathrm{0}\: \\ $$$${also}\:{we}\:{can}\:{write}\:\:\int_{−\infty} ^{+\infty} \:\frac{{dx}}{{x}}\:={lim}_{\xi\rightarrow+\infty} \:\:\int_{−\xi} ^{\xi} \:\frac{{dx}}{{x}} \\ $$$$={lim}_{\xi\rightarrow+\infty} \:\:\:\left[{ln}\mid{x}\mid\right]_{−\xi} ^{\xi} \:=\mathrm{0} \\ $$

Answered by mind is power last updated on 07/Apr/20

PV(∫_(−∞) ^(+∞) (1/x)dx)=0  cauchy  principal value  but ∫_(−∞) ^∞ (dx/x) not exist Riemann

$${PV}\left(\int_{−\infty} ^{+\infty} \frac{\mathrm{1}}{{x}}{dx}\right)=\mathrm{0} \\ $$$${cauchy}\:\:{principal}\:{value} \\ $$$${but}\:\int_{−\infty} ^{\infty} \frac{{dx}}{{x}}\:{not}\:{exist}\:{Riemann} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com