Reverse Product RuleHow did this happen. The guy goes "and here's the trick" and writes the result. How can he just look at that LHS and do what he did? What's the trick I'm missing?
differentiate top wrt x and u get those terms out, once u understand that in future DE's u can get ur int factor and skip a few stepsHow did this happen. The guy goes "and here's the trick" and writes the result. How can he just look at that LHS and do what he did? What's the trick I'm missing?
It doesn't. We need at least three vectors to span R3.
The question
Why does {v1, v2} still form a spanning set for R3 again?
I see no useful consequence of multiplying top and bottom by x. Also, you cannot bring the x out of the integral because it's a variable and not a constant.Why can't you just multiply top and bottom by x (bringing the top x out of the integral), thus cancelling the x on the top, and then do the tan sub? Rather than what they do? (obviously you get different answers tho)
oh thanksI see no useful consequence of multiplying top and bottom by x. Also, you cannot bring the x out of the integral because it's a variable and not a constant.
Multiplying top and bottom by x
does nothing. Their method is already efficient. Doing integration by substitution is what you want to avoid.
Add 0, multiply by 1. That's a key principle in algebraic manipulation. The trick is knowing the exact expression for 0 and 1, which only comes from experience or intuition.
splits up the fraction which when u integrate gives ur ln and arctan term
The third line explains why he does it. How he saw this is a result of foresight gain from experience/intuition as Paradoxica described.
following on from this,
1. The arithmetic mean of any two different real numbers strictly lies between them.Is there a trick to use that's not induction?
 Show that }a_n>a_{n+1}>b_{n+1}>b_n)
