Estimating the integral of a decreasing trigonometric function

[TehAdzMan]

Hi,

So this is part of an assignment for my numerical analysis class.

The integral is this:

$$<br /> \int_0^{\infty} e^{-x} \cos^2 (x^2) dx<br />$$

We are instructed to evaluate the integral from 0 to some large A using numerical methods (which I'm fine with), and then estimate the tail, ie the integral from A to infinity.

Basically we need to come up with some method to estimate and bound the remainder in the tail.
My idea was to substitute $$u = x^2$$ which gives $$\int_A^{\infty} \frac{e^{- \sqrt{u}} \cos^2(u)}{\sqrt{u}} du$$
and then I guess take the maximum of the cos function as 1 and just remove it, and use the remaining function, which I think would be quite easy to integrate and would give a maximum bound.
Any thoughts greatly appreciated.

Regards,
Adam

 

[Dick]

Hi,

So this is part of an assignment for my numerical analysis class.

The integral is this:

$$<br /> \int_0^{\infty} e^{-x} \cos^2 (x^2) dx<br />$$

We are instructed to evaluate the integral from 0 to some large A using numerical methods (which I'm fine with), and then estimate the tail, ie the integral from A to infinity.

Basically we need to come up with some method to estimate and bound the remainder in the tail.
My idea was to substitute $$u = x^2$$ which gives $$\int_A^{\infty} \frac{e^{- \sqrt{u}} \cos^2(u)}{\sqrt{u}} du$$
and then I guess take the maximum of the cos function as 1 and just remove it, and use the remaining function, which I think would be quite easy to integrate and would give a maximum bound.
Any thoughts greatly appreciated.

Regards,
Adam

Homework Statement

Homework Equations

The Attempt at a Solution

I don't see why you just don't use cos(x^2)^2<=1 to get a bound on the tail. But sure, you can do it that way too.

 

[TehAdzMan]

Yeah I just realized that my way is retarded and ends up with the exact same result.

Thanks!