Integration by parts computation

[the_kid]

Homework Statement

Consider the following integral:

I=\int^{\pi/4}_{0}cos(xt^{2})tan^{2}(t)dt

I'm trying to compute as many terms as possible of its asymptotic expansion as x\rightarrow\infty.

Homework Equations

x

The Attempt at a Solution

Let u=cos(xt^{2}). And dv=tan^{2}(t)dt.
Then du=-2xtcos(xt^{2})dt and v=\inttan^{2}(t)dt=tan(t)-t+C.

Integration by parts yields:

I=cos(xt^{2})tan(t)-tcos(xt^{2})+\int[2xtcos(xt^{2})tan(t)-2xt^{2}cos(xt^{2})]dt,
where all terms are evaluated from 0 to pi/4, obviously.

This feels wrong to me. Can anyone give me some help?

 

[tiny-tim]

hi the_kid!

(try using the X² button just above the Reply box )

Let u=cos(xt^{2}). And dv=tan^{2}(t)dt.
Then du=-2xtcos(xt^{2})dt

erm …

du=-2xtsin(xt²)dt ​