calculus question

To solve problems such as these you must resort to the substitution method. Make a new variable and call it u, then try to arrange the equation so that it looks like dy = (some integrable function of u) du. In this case, try calling x^2 u. You know that du = 2x. Try writing dy in terms of functions of u (you are still going to be using e ^ (something), and see if you can finish it up.
Brendan

With e^u, the integral almost always contains e^u. Try deriving it and see what you come up with. I find it easier to "guess" integrals than to do it formally with substitutions.

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Cédric

It might help... Integration by parts

[edited by - bloodscourge on December 13, 2002 12:45:28 PM]

Also, always be on the lookout for integrals in the form ∫u*du dx, where u is a function of x.

Use integration by parts


integral(f(x)*g''(x)) = f(x)*g(x) - integral(f''(x)*g(x))

It has absolutely nothing to do with this problem, don't even bother with it. You use int by parts for things like x*ln x or x*cosx. And, I would not reccomend blindly guessing what the intergral is, that isn't a skill, its luck. Learn the method, love the method, use the method, because it is used all over the place.

Change:
This is just my opinion. When I took APBC calc, I learned the techniques, and then learned the types of problems to apply them on. I don't support writing out what u and du is every time you do a problem, because with practice you learn that. But I wouldn't say guessing is too much of the problem. You see roots with x^2 and something else, you are going to do trig substitution, you know when to do it by parts, and when to do a u-substitution.

Brendan

[edited by - Punty50 on December 13, 2002 9:53:05 PM]

quote:

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Original post by Punty50
And, I would not reccomend blindly guessing what the intergral is, that isn''t a skill, its luck. Learn the method, love the method, use the method, because it is used all over the place.

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I wasn''t very good at looking for primitives, but I think there is no such thing as "the method". There are several methods that you can try, and if you they don''t give you a solution... well, maybe there isn''t any. It is probably more important to learn numeric methods, at least for game developing.

"Blindly guessing what the integral is" is not such a bad idea. Actually, trying to do that, you may develop some intuitions on how to solve those problems.

I guess we have very different experiences with math.

quote:

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And, I would not reccomend blindly guessing what the intergral is.

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Poor choice of words, since it''s hardly "blindly guessing the integral". If anything, if you can see or know a pattern in the integral (such as the derivative of e^x is e^x, you can get an idea as to what the solution might look like and work from there. Like AP said, methods of integration don''t rely on a fixed system like differentiation (one reason why they make a distinction between integral and differential calculus). Any information you can gather from just observation is helpful, especially went you are integrating non-trivial equations.

I''ll second or third the comment that finding a solution to problems such as this is not a matter of "blindly guessing the integral." As Anonymous Poster said, integration is like an art and to do it well you will develop an eye for it. AP obviously has somewhat of an eye for doing integration, .

I don''t think this discussion has been too harmful to amish1234''s education; however, in terms of future homework questions, I do wish him/her luck with either obtaining help from the teacher or successfully completing the brutal rampage!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Latest AP:
This problem does have classical methods of solving it, specifically, u-substitution. if u = x^2, then du = 2x and the integral now become (int) .5*e^u*du, which we evaluate as .5e^u which is .5*e^x. As for my comments, sorry. I personally can look at a problem and see what sort of integration process is necessary. Is that guessing? Perhaps. The one point that I was trying to make is that you need to have some sort of idea as to what to look for, otherwise you are going to be very lost. So, if you can identify what process you need, or if the problem is similar to one you have encountered, than guess, because chances are you are going to be right. This is not a blind guess, but rather an educated one, which I think you will do at some point. And finally, don''t get something like a TI-89 to do your work. You will fail soooo miserably in math because you will forget some basic techniques. This happened to my best friend, and it was a real shame. Use it to check homework and answers, because its features are very powerful and useful, but don''t let it do the work for you.
Brendan

Use Taylor series

Does anyone know which method programs like derive or the TI-89 use? Do they solve all analytically solveable integrals?

GA

I imagine calculators have a limited collection of closed-form integrals built-in, and wouldn''t be able to solve them all. (Heck, they''re probably not all even known.)

I wonder if they have implemented the source code for Maple V in a chip?

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Here is a link to how mathmatica''s Integration functions work. It is extremely interesting and intriguing.

http://integrals.wolfram.com/about/inside/

I would imagine that the TI calc''s can''t do stuff this sophisticated, so they probably have general algorithms that can solve some of them. If you look at a "brief" integral table, all of those functions can be done, as well as things that are done recursively when integrating by parts. Its all really neat stuff to look into.

Brendan

TI-89+ are usually really good about finding an integral. Notice how I say an integral. You see, they just don''t have the human touch. They''ll find an integral, but it will usually be 8 times as complex and 56 terms longer than what a human could find I wouldn''t complain, they do it all in a few seconds