Explained: "Find the integral of sin⁡(x2)"

elvishwitchxyp

Answered question

2021-12-10

Find the integral of

\sin (x^{2})

recoronarrv

Beginner2021-12-11Added 20 answers

Substitute

u = \frac{\sqrt{2} \times x}{\sqrt{π}}

d x = \frac{\sqrt{π}}{\sqrt{2}} \times d u

\int \sin (x^{2}) \times d x

= \frac{\sqrt{π}}{\sqrt{2}} \int \sin (\frac{π \times u^{2}}{2}) \times d u

This is special integral Fresnel integral =S(u)

\frac{π}{2} \int \sin (\frac{π \times u^{2}}{2}) \times d u = \frac{\sqrt{π} \times S (u)}{\sqrt{2}}

= \frac{\sqrt{π} \times S (\frac{\sqrt{2} \times x}{\sqrt{π}})}{\sqrt{2}} + C

if we undo the substitution.

Mary Herrera

Beginner2021-12-12Added 37 answers

\int \sin (x^{2}) d x = \sqrt{\frac{π}{2}} S (\sqrt{\frac{2}{π}} x) = \sqrt{\frac{π}{2}} S (\sqrt{\frac{2}{π}} x) + C

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