# Evaluating the integral $\int_{0}^{2} \sin(x^2*\pi) dx$

I have no idea how to do this integral:

$$\int_{0}^{2} \sin(x^2\pi) dx$$

Can you please tell me how to calculate the definite integral? thanks

• It is not possible to get an elementary function for this integral. Maple gives an answer that contains a Fresnel function. – Bernard Massé Sep 27 '18 at 23:46
• Use a numerical approximation method – MPW Sep 28 '18 at 0:01

$$\int_{0}^{2}\sin\left(\pi x^2\right)\,dx =\frac{1}{\sqrt{2}}\int_{0}^{2\sqrt{2}}\sin\left(\frac{\pi}{2}x^2\right)\,dx =\color{red}{\frac{S\left(2\sqrt{2}\right)}{\sqrt{2}}} \approx 0.2743355$$ Fresnel Integral.
note that: $$\int_0^2\sin(x^2\pi)dx=\frac{1}{\sqrt{\pi}}\int_0^{2\sqrt{\pi}}\sin(u^2)du=\frac{1}{\sqrt{\pi}}\int_0^{2\sqrt{\pi}}\sum_{n=0}^\infty(-1)^n\frac{x^{4n+2}}{(2n+1)!}dx$$ and this can easily be expressed as a series