Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

LOL; As you can see our ungrateful kids have cr维pped the whole math category to frustrating boredom! As a change I propose a bit more interesting stuff.

1) this problem was posted 2 weeks ago

http://answers.yahoo.com/question/index;...

I wonder who can do it besides Scythian;

evaluate 鈭玠x sin(x*2007^207) /sin(x) {x=0 until pi};

2) here I shall not mention the source as it contains solution;

evaluate 鈭玠x (cos x)^2 /(1+ exp(sinx)) {for x from 鈥損i/2 to +pi/2};

Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

Well, Scythian gave the answer to the first integral as 蟺.

He was right, but never showed how he got the answer.

We will show, more generally, that

鈭?0..蟺) sin(2n+1)x dx/ sin x = 蟺 for n any positive integer.

I browsed through my trig book(Trigonometry,

Functions and Applications, by Paul Foerster, p.146)

and found an outline of a proof of the following identity:

1+ cos x + cos 2x + ... + cos nx = 陆(1+sin(2n+1)x /2 / sin(x/2) )

Replacing x by 2x in this identity and doubling both sides,

we get

1 + 2(cos 2x + cos 4x + ... + cos 2nx )= sin(2n+1)x/ sin(x)

So,

鈭?0..蟺) sin(2n+1)x/ sin x dx=

(x +2( sin 2x/2 + sin 4x/4 + ... + sin 2n x/2n))(0..蟺)

which just equals x(0..蟺) = 蟺.

I haven't made any headway on the second integral yet.

I tried using substitution u = sin x

and parts, letting u = cos x and v = cos x/(1+ e^sin x),

but these just lead to dead ends.

I did run it through the link below and

it gave the answer as 蟺/4, but I have no idea

how to prove it right now. Will keep trying!

Yes, let's keep posting these challenging

problems. I enjoy them!

Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

For the second one consider the functions

f(x) = cos^2 (x) / (1 + e^sin(x))

and g(x) = cos^2 (x) / (1 + e^{-sin(x)})

f(-x) = g(x)

This means that

I = 鈭玣(x) dx = 鈭玤(x) dx

where both integrals go from -蟺/2 to 蟺/2 Report It

Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

Now 2I = 鈭玔f(x) + g(x)] dx

= 鈭?cos^2(x) * [1/(1+e^sinx) + 1/(1+e^(-sinx))]dx

=鈭玞os^2(x)*[2+e^sinx+e^-sinx] /[2+e^sinx+e^-sinx]dx

=鈭玞os^2(x) dx = 蟺/2

I = 蟺/4 Report It

Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

-I鈥檓 shot down! No comments; I peeked it there

http://answers.yahoo.com/quest...

and just made it up a bit to disguise from search option! Thank you Doc! Report It

Integration; big heads!!! plzzzzz heeeeelp, wailing and pulling my hair out?

what use a calculator