Can the sum of two divergent integrals yield a convergent integral? Given:
$f(x) + g(x) \neq 0$
$\int_a^b f(x) dx $ diverges
$\int_a^b g(x) dx $ diverges
$a$ and $b$ can be real numbers or $\pm\infty$
Find $f$ and $g$ such that:
$$\int_a^b (f(x)+g(x)) dx = L$$
for some finite $L$.
I really can't seem to think of any examples asides from $f+g=0$. Is there a solution to this at all?

Suppose $\int_a^{b} f(x)dx $ diverges and $\int_a^{b} h(x)dx$ converges with $h \neq 0$. Take $g=h-f$
$$\int_a^b (f(x)+g(x)) dx =\int_a^{b} h(x)dx$$ is convergent.
Also, $\int g$ is divergent.
Example: $f(x)=\frac  1 {x-a}, h(x)=1$.
