Estimating the integral of a decreasing trigonometric function

1. Nov 21, 2013

Hi,

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

The integral is this:

$$\int_0^{\infty} e^{-x} \cos^2 (x^2) dx$$

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,
1. The problem statement, all variables and given/known data

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3. The attempt at a solution

2. Nov 21, 2013

Dick

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.

Last edited: Nov 21, 2013
3. Nov 21, 2013