- #1

sarrah1

- 66

- 0

This is a very very simple question

I can show when $f$ is integrable and is even i.e. $f(-x)=f(x)$ then

$\int_{-a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$

what about improper integrals of even functions, like the function ${x}^{2}\ln\left| x \right|$ this function is even but undefined at the origin. Yet it has a limit equal to zero there. I wish to integrate it from $[-1,1]$. I know that I can partition the integral into $[-1,0]$ then $[0,1]$ and in each interval I can carry on the improper integration. My simple question is since the function is not Riemann integrable, but Cauchy integrable can I still write

$\int_{-1}^{1} \,{x}^{2}\ln\left| x \right|\,dx=2\int_{0}^{1} \,{x}^{2}\ln\left| x \right|\,dx=2\lim_{{c}\to{{0}^{+}}}\int_{c}^{1} \,{x}^{2}\ln\left| x \right|\,dx$

Many thanks

Sarrah