How do I find the integral of
F(x) = the integral of 1/1+t^2 dt from 0 to x + the integral of 1/1+t^2 dt from 0 to 1/x?
and how do I determine that F(x) is constant on (-infinity,0) and constant on (0,infinity)?
Any help appreciated!
Thanks
How do I find the integral of
F(x) = the integral of 1/1+t^2 dt from 0 to x + the integral of 1/1+t^2 dt from 0 to 1/x?
and how do I determine that F(x) is constant on (-infinity,0) and constant on (0,infinity)?
Any help appreciated!
Thanks
$\displaystyle F(x)=\int\limits_0^x\frac{dt}{1+t^2}+\int\limits_0 ^{1\slash x}\frac{dt}{1+t^2}=\left[\arctan t\right]_0^x+\left[\arctan t\right]_0^{1\slash x}=\arctan x+\arctan\left(1\slash x\right)$.
To determine that this function is a constant in some interval just differentiate it...what should you get?
Tonio
Since (F+ G)'= F+ G', you could also show this function is a constant by differentiating each of the integrals separately and adding. By the "Fundamental Theorem of Calculus" the derivative of the first integral is just $\displaystyle \frac{1}{1+ x^2}$. By the "Fundamental Theorem of Calculus", together with the chain rule, the derivative of the second integral is $\displaystyle \frac{1}{1+ \frac{1}{x^2}}\left(\frac{-1}{x^2}\right)= -\frac{1}{1+ x^2}$. Those clearly add to 0, for all x (except x= 0 where $\displaystyle \frac{1}{1+ \frac{1}{x^2}}$ is not defined), so the function is a constant.