JimmyP
10-22-2009, 12:57 AM
I've searched high and low and can't seem to find a list of the coolest integrals ever, so I've decided to ask for some help in collecting a few.
Since one can't just go about posting any integral one pleases I think we should perhaps define what a cooleset integral ever is. These should be integrals for calc I and II students who are interested in learning more about the subtleties of calc.
A coolest integral ever could do some of the following (just suggestions):
0. First and foremost a coolest integral ever should excite the student. They should be able to smile and say, "Hey, that is kinda cool!" It should promote their interest in mathematics and provoke their curiosity of the subtleties of math.
1. A coolest integral ever should be able to be done by first or second year calc students.This means techniques like creative u-substitutions, integration by parts, sweet trig substitutions, improper integrals, etc.
2. Use some kind of really cool, or really sneaky symbolic manipulation to make the integral doable or at least a little easier. The symbolic manipulation could be designed to use some of those forgotten rules from algebra, say rules of exponents, or perhaps rules of logarithms, trig identities, etc. Also, multiplying/dividing by one is pretty sweet.
3. It could be part of a more involved problem, say a physics problem, etc, but it should still be mostly about the math.
4. Anything else you deem worthy of a coolest integral ever, but remember, it shouldn't be an integral which goes off on some crazy integration by parts tangent where you loose minus signs, use up your eraser and end up having to go to Office Depot for more paper when you're done with the whole ordeal. Unless it's really worth it, and it might just be!
So here is my entry and an example:
int[e^((e^x)+x)]
Integral of e raised to the sum of e to the x plus x.
This one is cool because it falls apart with simple rules of exponents and then an easy u-sub. It really is a good one for calc I kids!
Since one can't just go about posting any integral one pleases I think we should perhaps define what a cooleset integral ever is. These should be integrals for calc I and II students who are interested in learning more about the subtleties of calc.
A coolest integral ever could do some of the following (just suggestions):
0. First and foremost a coolest integral ever should excite the student. They should be able to smile and say, "Hey, that is kinda cool!" It should promote their interest in mathematics and provoke their curiosity of the subtleties of math.
1. A coolest integral ever should be able to be done by first or second year calc students.This means techniques like creative u-substitutions, integration by parts, sweet trig substitutions, improper integrals, etc.
2. Use some kind of really cool, or really sneaky symbolic manipulation to make the integral doable or at least a little easier. The symbolic manipulation could be designed to use some of those forgotten rules from algebra, say rules of exponents, or perhaps rules of logarithms, trig identities, etc. Also, multiplying/dividing by one is pretty sweet.
3. It could be part of a more involved problem, say a physics problem, etc, but it should still be mostly about the math.
4. Anything else you deem worthy of a coolest integral ever, but remember, it shouldn't be an integral which goes off on some crazy integration by parts tangent where you loose minus signs, use up your eraser and end up having to go to Office Depot for more paper when you're done with the whole ordeal. Unless it's really worth it, and it might just be!
So here is my entry and an example:
int[e^((e^x)+x)]
Integral of e raised to the sum of e to the x plus x.
This one is cool because it falls apart with simple rules of exponents and then an easy u-sub. It really is a good one for calc I kids!