the world of integrating by parts.

"If numbers aren't beautiful - I don't know what is" - Paul Erdos

So anyway, my first pattern my teacher wishes for me to notice, while exploring that sea of number lore in private, is how to do integrals of two functions. It's really the same pattern as the derivative of two things multiplied together, only now you're just doing the integral of the thing.

Really it's sorta intereting - I just need to plug away, another interesting thing is the relation between the sin and arc sin, which shows how the graph looks when you just replace the x and y's of each function.

Things

" What would really be a lesson that everybody should learn is that things come out of nothing - things evolve out of nothing. The tiniest seed in the right situation turns into the most beautiful forest, and then, the most promising seed in the wrong situation turns into nothing...
it gives people confidence.. that's how things work.. you can say.. 'Well, I know that things come from nothing very much and start from unpromising beginnings, and I'm an unpromising beginnigng - I could start something"

Mathematics is not so much ability as attitude

..the renee tape is one of his favaroites because of how beautifully it illustrates what he considers to be the secret to learning mathematics: Twenty-two minutes pass from the moment she says, "Ahhhh. That means something now." That's a long time. "If I put the average person in the same situation.. I'm guessing that after the first few attempts, they would have said, "I don't get it. I need you to explain it."..Schoenfield once asked a group of.. students how long they would work on a homework question before they concluded it wasn't theirs to solve.. the average was two minutes. But [she] persists. She experiments. She goes back over the same
issues time and again. She thinks out loud. She keeps going and going.
She won't give up.

"Oh! it's... " Her facelights up... Ohhh okay. Now I see.. That means
something"

He teaches a course.. I pick a problem that I don't know how to solve, 'he says' I tell my students, 'You're going to have a two-week take-home exam [or in my case.. 6 days to do a homework assignment]..' I know your habits. You're going to do nothing for the first week and start it next week, and I Want to warn you now: If you only spend one week on this, you're not going to solve it. If, on the other hand, you start working on it the minute you get the assignent,
[it might not work the first second.. but by week two [or in my case day 5] you'll have solved it.

Mathematics is not so much ability as attitude. Success is a function of persistence and doggedness and the willingness to work hard for twenty two minutes to make sense of something that most people would give up on after thirty seconds.