Please help integrate ∫ [0..+inf] sin(x²) dx?

Question

Please help me people... it's my elder brother's homework...
unless I do it he will tell my mom what websites I browsed yesterday.

Answer 1

(1/2)√(π/2) is the answer.

The integral equates to:

√(π/2) * S[x√(2/π)]
Where S[x] is the Fresnel integral S
It is known that:
S[0] = 0
S[∞] = 1/2
which yields the solution given.

(1/2)√(π/2) = √(2π) / 4 = .626657....

Answer 2

What websites did you go to? That's a tough one.

If you go to

http://integrals.wolfram.com/index.jsp

And type in "sin(x^2)" and hit the button, you'll see that the antiderivative is:

√(π/2) S(√2x/π)

Where S represents the "Fresnel integral":

http://mathworld.wolfram.com/FresnelIntegrals.html

I'm not familiar with Fresnel integrals, but just looking at that thing, it appears that it converges toward 0.5 as x goes to infinity.

So I think the answer would be:

√(π/2) (1/2) = √(2π)/4 or 0.25√(2π)

Answer 3

∫ (0 to infinity, sin(x²) dx)

Since this is an improper integral, we must first change this into a limit.

lim (0 to t, sin(x²) dx)
t → ∞

You must understand that sin(x²) is a difficult integral to integrate; the answer cannot be expressed in terms of elementary functions. Nevertheless, this integral diverges and you should get an answer of "does not exist."

Edit: Judging by the answers below, I stand corrected and am wrong.