Help with integrals (this one could take a while to solve so...)?

Question

I don't expect anyone to solve this for me, but my hope is that you can (at least) guide me a little. I'm a first year student so I don't have a lot of calculus knowledge and tend to struggle with integration concepts involving power series representations, trigonometric substitutions,...etc. I really don't think that I have enough background to solve the following, but I have the chance to earn myself $2,000 (from my uncle who's a math wizard to say the least) if I can successfully solve (showing all the necessary steps) the following indefinite integral (s):


s {[x^2]/[1 + sin(x)] dx}

I already discovered the answer at: http://integrals.wolfram.com/index.jsp, but I have no clue what the induction steps are, or how my uncle could have solved this monster on his own. I would be so (...so,so,so...) grateful if someone can help me out here.

Thanks a lot

Answer 1

IF you want to try and get it in that EXACT form dicatated by Mathematica, I suggest you start with u=1+sin(x) and then solve for x in terms of u and do some resubstitution. You'll get the arcsin function, which becomes a polynomial in terms of another dummy variable when you start the first u*dv substitution. Notice I said "FIRST", because to get it in the form that you want, you'll have to do it about 6-10 times. Also, you'll have to do some "right triangle simplification" throughout the problem as well. Of course, then you get to factor it. Have at it. I did this very thing, stopping at the 3rd u*dv substitution because it is just ridiculous.
I did, however, get several of the terms. I think if you decide to do this, you're going to need lots of paper. I was doing it symbollically in a CAD program to keep things organized and such, and it was already 5 pages. Oh, and not to mention that this "answer" given by Mathematica isn't even a closed form "answer". It left it in terms of a polylogarithm, which is still an integral hidden in notation.

The point is that the fuction (among others) is best integrated numerically because there is no useful closed form for this function. This is what I did and it took me 2 minutes.

x^2/(1+sin(x))=x^2 - x^3 + x^4 - (5/6)x^5 + (2/3)x^6 - (61/120)x^7 + (17/45)x^8 - (277/1008)x^9 .................

Integrate termwise, which is fricking easy.

(1/3)x^3 - (1/4)x^4 + (1/5)x^5 - (5/36)x^6 + (2/21)x^7......

Notice it is not undefined (like the original function) and the error term goes to 10^-7 after the first 5 terms. I'd say that's as accurate as anyone would ever really need to be, and if not, they can always expand this function further.

Tell your uncle he's a bastard just for me. He's teasing you.

Answer 2

Doesn't this thing get evil when sin(x) hits minus 1 and makes that denominator go to zero? I tried typing the thing into MAPLE but all I've got is a demo version and it gagged on it, saying something cryptic about int/treduce...

The first thing I'd try is changing variables using u = sin(x). See where that gets you.

Answer 3

I have no clue. I can figure out the derivative of that, but other than that, I'm afriad I can't help you.

I wish you success though as you continue your endeavor.

Answer 4

First thing: put down that damn C. I missed many questions because of damned C. FU C.

Answer 5

How much of the $2K do I get? I'm not a chump, you know.