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#