0 to 1/2∫(sin(x^2))dx

Answer 1

Recall that
sin^2 x = 1/2 - (1/2)cos (2x)
so...
∫(sin(x^2))dx from 0 to 1/2
= ∫(1/2 - (1/2)cos (2x) dx from 0 to 1/2

= x/2 - (1/4)sin (2x) from 0 to 1/2
= ......

Answer 2

you cannot integrate sin(x^2) using elementary functions.
Some numeric method is needed, or one of the special functions defined as integrals. In this case, the Fresnel Integral.