- #1

- 1,247

- 11

## Homework Statement

Integrating ##^{\frac{\pi}{2}}\int_{\frac{-\pi}{2}}(1-u^{2})^\frac{1}{2}u_{x}dx##, and using the result : ##\int(1-u^{2})^{\frac{1}{2}}=\frac{1}{2}u(1-u^{2})^{\frac{1}{2}}+\frac{1}{2}arcsin(u)##

## Homework Equations

I'm pretty sure it is just the integral itself were I am going to wrong. But to provide context, just in case, it is a bounding the energy question using the Bogomolnyi argument from a soliton course. Here's the two lines, I am just stuck on how we get from the top line to the last, so the 2nd term in both lines.

##E \geq \frac{1}{2}(u_{x}\pm(1-u^{2})^{\frac{1}{2}})^{2}dx \mp (1-u^{2})^\frac{1}{2} u_{x} dx ##

## = \frac{1}{2}(u_{x}\pm(1-u^{2})^{\frac{1}{2}})^{2}dx \mp [\frac{1}{2}u(1-u^{2})^{\frac{1}{2}}+\frac{1}{2}arcsin(u)]^{\frac{\pi}{2}}_{\frac{-\pi}{2}} ##

## The Attempt at a Solution

Using ##dx=dx/du * du ## I see how we can replace ##u_{x} dx## with ##du## and so this allows us to use the result provided, BUT surely then the limits too need changing - to ##\pm 1 ##?

**Thanks in advance.**

Last edited: