- #1
parton
- 79
- 1
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.
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.
