Question

Integral of sin(x^2)

Answer 1

$\smallint{sin(x^2)}.dx$

Substitute $u=\frac{\sqrt{2}.x}{\sqrt{π}}$

$dx=\frac{\sqrt{π}}{\sqrt{2}}.du$

$\smallint{sin(x^2)}.dx= \frac{\sqrt{π}}{\sqrt{2}}\smallint{sin(\frac{π.u^2}{2})}.du$

These is special integral Fresnel integral $=S(u)$

Plug in solved integrals

$\frac{\sqrt{π}}{\sqrt{2}}\smallint{sin(\frac{π.u^2}{2})}.du=\frac{\sqrt{π}.S(u)}{\sqrt2}$

Undo Substitution $u=\frac{\sqrt{2}.x}{\sqrt{π}}$

$=\frac{\sqrt{π}.S( \frac{\sqrt{2}.x}{\sqrt{π}} )}{\sqrt2}+c$

Hence

$\smallint{sin(x^2)}.dx =\frac{\sqrt{π}.S( \frac{\sqrt{2}.x}{\sqrt{π}} )}{\sqrt2}+c$