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If I have two functions $f$ and $g$ and aim at computing: $$I:=\int (f-g) = \int f - \int g$$ in some region and I find out: $$\int f = \int g = +\infty.$$

Can I conclude that $I=\infty$? Or is there a possibility for "cancellation" of infinities? I got a bit confused at this point.

Thanks for any help! :)

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    Not necessarily. Trivially you can take $f = g$. Like $f(x) = g(x) = 1/x$ where the integral is over $[0,+\infty)$. I'll post an actual answer if I can come up with a nontrivial example later but someone else will likely get there first. – 2017-02-02
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    Even simpler, take $f = g = 1$, the constant functions. – 2017-02-02

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if $f(t)=\frac1t$ and $g(t)=\frac{(1-t)^n}{t}$ then $$ \int_0^1 (f(t)-g(t)) dt = H_n = 1+\frac12+\frac13 + \dots + \frac1n $$ but the integrals taken separately diverge

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    That's a really cool integral. – 2017-02-02