Problem with divergent integral

[parton]

I'm confused with the following integral.

Let a > 1.

$$\int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left[ \log \left| \dfrac{t}{t-1} \right| \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right|$$

This should be the correct result. But I could also decompose the integral into two parts (because the integrand is a sum) and compute:

$$\int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left| \log \vert t \vert \right|_{a}^{\infty} - \left[ \log \vert t - 1 \vert \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| + \infty - \infty$$

But $$\infty - \infty$$ is of course not defined!

Where did I make a mistake? I don't find it.

 

[Bob_for_short]

Decomposing the integral into two parts is justified only if both integrals are finite.

 

[parton]

Thanks