How do you find the integral of $sin(x^2)$ ?

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Answer 1 · 2018-05-22T00:32:46

$color(blue)[intsin(x^2)*dx=(sqrtpi*S((sqrt2*x)/sqrtpi))/sqrt2+c]$

show the steps below:

Substitute $color(red)[u=(sqrt2*x)/sqrtpi$

$dx=sqrtpi/sqrt2*du$

$intsin(x^2)*dx$

$=sqrtpi/sqrt2intsin((pi*u^2)/2)*du$

These is special integral Fresnel integral

$=S(u)$

Plug in solved integrals:

$sqrtpi/sqrt2intsin((pi*u^2)/2)*du=(sqrtpi*S(u))/sqrt2$

Undo substitution $color(red)[u=(sqrt2*x)/sqrtpi$

$=(sqrtpi*S((sqrt2*x)/sqrtpi))/sqrt2+c$

How do you find the integral of sin(x^2)sin(x^2)?