∫sin (x²) dx??? (integral of sin(x^2))?

Question

I heard that sin (x²) is a Non-integrable function. What does that mean?

I tried solving it as follows:
put x² = t.
Hence, 2x dx = dt
dx = 1/2x dt
dx = 1/2√t dt

So,
∫sin (x²) dx = ∫(1/2√t) sin t dt
Which can be integrated by parts (twice) and resubstituting t = x². What's wrong in it?

Please help...
Thanks!

Anonymous · Accepted Answer

A non-integrable function simply means that the answer cannot be expressed as a combination of "simple" functions.

This integral is expressible in terms of a special function

Sqrt[pi/2]*Fresnels[x*Sqrt[2/pi]]   

I don't know what error you made evaluating you integral but clearly it cannot be correct.  Sin[x^2] oscillates more and more rapidly between -1 and 1 as x increases, so clearly it cannot grow monotonically as your solution does.

sahsjing · Answer

It means you don't have analytical solution or elementary solution for â«sin (xÂ²) dx.

By the way, integration by parts doesn't work here. If you post your solution, I can show you where you did wrong.

Joe L · Answer

Apparently, you need what's called a Fresnel integral to solve it.

Go to this website and plug in your integral:

http://integrals.wolfram.com/index.jsp

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∫sin (x²) dx??? (integral of sin(x^2))?

3 Answers