intergration help!?

Question

i need to find intergral of

sin(t^2 - 5)

I tried to substitute u = sqrt(t^2 - 5)
So u^2 = t^2 - 5
And du/dt = blah blah = t/u
Thus dt = u*du / t

Now S( sin(t^2-5) * dt )
= S (u *sin(u^2) / t * du)

And.....i'm stuck. Please help!

Answer 1

Two problems here.
1). Your substitution is not done correctly.
Better to let u = t², t = √u du, dt = 1/(2√u) du
Now we get
∫ ½ sin(u-5) du/√u,
which is no easier to evaluate than before!
2). Your integral is nonelementary.
There is no elementary antiderivative for it.
It is an example of a Fresnel integral.
These integrals arise in the study of diffraction of light.
I just thought of another way to reduce it to
standard Fresnel form:
sin(t²-5) = sin t² cos 5 - cos t² sin 5.
Now split the integral in two and take
out the constants and you have the
difference of 2 standard Fresnel integrals.

Answer 2

This is a "classic" example of a function that does not have an explicit anti-derivaitve. Sure you got the question correct? Nobody can do this -- not possible!
sorry.

If you have limits of integration and it is a definite integral, then we can evaluate the integral numerically...(approxiate value)

Answer 3

it can't be calculated ,cause it's a continues integral...