Please help me integrate: sin x^2 dx?

Question

Is this integrable? Please help me... I need to explain it clearly to my teacher so I can pass the subject... :)

This is: sin (x)^2 NOT (sin x)^2.
The exponent 2 takes effect on x only.

Answer 1

Express (sin x)^2 in terms of cos 2x,
using cos(2x) = 1-2sin^2(x). It is then straightforward.

Answer 2

This is an example of a Fresnel integral. It
cannot be integrated in terms of elementary
functions. Let's change its form slightly to
get another nonelementary integral:
Let u = x² , x = √u, dx = du / (2√u)
So we get ∫sin(u) du/ 2√u,
which is also nonelementary.
Fresnel integrals are found in the study of diffraction
of light.
You might also Google cornu spirals for
another application of Fresnel integrals.
Good luck with your course!

Answer 3

u must know this trigonometric relation

cos 2x = 1 - 2 sin^2 x

= 2 cos^2 x - 1

so when u hav sin^2 x or cos^2 x term replace them in terms of cos 2x using the above relations

now

sin^2 x = (1 - cos 2x) / 2

integrating with respect to x

ans ={ x - (sin 2x)/2 } /2

= x/2 - (sin 2x)/4

Answer 4

To solve these type of integrals you need to use "Fresnel Integrals". Check the source links below,