Question

Show that

⁰∫[infinity] sin(x²)dx = ⁰∫[infinity] cos(x²) dx =√2π/2
These are the Fresnel integrals. Here, ⁰∫[infinity] sin(x²)dx denotes the integral of sin(x²) over the interval from 0 to [infinity], and ⁰∫[infinity] cos(x²) dx denotes the integral of cos(x²) over the same interval.

Answer 1

To show that the integrals ⁰∫[infinity] sin(x²)dx and ⁰∫[infinity] cos(x²) dx both equal √2π/2, we can use the Fresnel integrals and wire symmetry. By expressing r and sin in terms of x and R, we can evaluate the integral or use the fact that the average over a complete cycle for cos² (5) is the same as for sin² (5).

Step-by-step explanation:

To show that ⁰∫[infinity] sin(x²)dx = ⁰∫[infinity] cos(x²) dx = √2π/2, we can use the Fresnel integrals. The wire symmetry about point O allows us to integrate from 0 to infinity and double the answer. We can write expressions for r and sin in terms of x and R. Next, we can evaluate the integral or note that the average over a complete cycle for cos² (5) is the same as for sin² (5). This leads us to the desired result √2π/2.