- #1
Mr Davis 97
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I am asked to evaluate the following integral: ##\displaystyle \int_0^{ \infty} \log \left(1+ \frac{a^2}{x^2} \right) dx##. So of course this is an improper integral, but I am confused about how to go writing out the integral. From previous courses, I know that you should split the integral so that you have two clear improper integrals, rather than one that is doubly improper, so something like ##\displaystyle \int_0^1 \log \left(1+ \frac{a^2}{x^2} \right) dx + \displaystyle \int_1^{ \infty} \log \left(1+ \frac{a^2}{x^2} \right) dx##, and then we're supposed to write this out with limits: ##\displaystyle \lim_{t \rightarrow 0} \int_t^1 \log \left(1+ \frac{a^2}{x^2} \right) dx + \lim_{s \rightarrow \infty} \int_1^s \log \left(1+ \frac{a^2}{x^2} \right) dx##, and we evaluate each improper integral separately. This is how I learned to do it, but it all seems very cumbersome.
Is there a better way of doing this? I feel like doing it this way just takes an unnecessarily long time
Is there a better way of doing this? I feel like doing it this way just takes an unnecessarily long time
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