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=$\int$tan$^{2}$(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 ​