TITLE: How do I integrate $\int_{0}^{1}\!\sin x^2\,dx$?
QUESTION [5 upvotes]: How do I integrate 
$$
\int_{0}^{1}\!\sin x^2\,dx?
$$
Will it be so complicated?

REPLY [11 votes]: This integral is known as the Fresnel Integral, defined by:
$$S(x)=\int_0^x \sin(t^2) \, dt$$ 
And cannot be evaluated without the use of a series approximation.
Numerically, this is around $0.31$.