Computing a double integral $\int_{-\infty}^\infty\int_{-\infty}^\infty\frac{f(t)}{1+{(x+g(t))}^2}dt\ dx$

Question

Let $f,g$ be continuous, with $f$ integrable.

How can one evaluate $\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{f(t)}{1+{(x+g(t))}^2}dt\ dx$ ?

Any hint would be welcome.

I have not learned Funibi/Tonelli for indefinite integrals yet.

PS : To those who are voting to close as "too broad", would you mind explaining what is too broad in my question ? I am just asking for a way to evaluate the integral, or a hint that would lead me to the solution.

Answer 1

$\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{f(t)}{1+{(x+g(t))}^2}dt~dx$

$=\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{f(t)}{1+{(x+g(t))}^2}dx~dt$

$=\int_{-\infty}^\infty\left[f(t)\tan^{-1}(x+g(t))\right]_{-\infty}^\infty~dt$

$=\pi\int_{-\infty}^\infty f(t)~dt$