Jim Coroneos' 100 Integrals ~ 022 ~ ∫√[(x+1)/(x-1)].dx

[@MrVoayer]

Dear Greame, when solving the first integral at 2:04 you say "...if I take the derivative of x squared minus 1..." What you are really talking about is "u substitution" (or "t substitution" as I was thought). Thus, I think, for pure educational purposes and to make things clearly visible, I think it would have been better if you had worked it out that way. In that case we should substitute u for (x squared minus 1) in the first integral. Then du = 2x dx ==> 1/2 du = x dx and we are good to go. We end up with an integral of u to the power of -1/2 which is easily solved by using "power rule". With a lot of integrals under your belt it becomes easy to see the whole substitution proces at once but for strictly pedagogical purposes I like to see it written. The second integral looks innocuous but is, in fact, challenging one. It deserves a video on its own.As always: thank you for teaching us !

[@CrystalClearMaths]

+MrVoayer Hello, again, friend. I certainly agree. Because I was running out of room on the whiteboard I opted for evaluating the integral in one step. This meant that I tried to explain the thought processes necessary, but it does not make for good videos or for good pedagogy. I should have used the substitution that you described. The second integral is, indeed, challenging. Again, for space reasons (and because it is recognised as a 'standard integral form'), I decided to appeal to the standard form here. I have plans to produce a separate video series (after this one) in which I will address many of the standard forms. There are often a number of ways in which to evaluate each integral. In order to show/demonstrate some of that variety, I do not always choose the 'best' or most conventional method for each video. This is, perhaps, a weakness ... but I am also using the opportunity to instruct students in alternative ways of setting their work out of thinking through the process. You have made an excellent point and I hope viewers will read your comment!