A clarification on a step in an integration question

1. Jan 16, 2016

mattyk

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

I was given this question as a part of an assignment and lost a mark because of a step.

2. Relevant equations
the integral of
cos^5(x) dx

after some fiddling and substitution it gets to this

(1 - u^2)^2 du
In the solutions there is a step that says
refine
= (u^2 - 1)^2
basically switching the the 1 and the u^2 around.

3. The attempt at a solution
Is this possible and if so how?

2. Jan 16, 2016

BvU

It is possible because $(-1)^2 = 1$ and the distributive property of multiplication:\eqalign { (a^2-1)^2 & = 1 * (a^2-1)^2 \\ & = (-1)^2 * (a^2-1)^2 \\ & = ( -1*(a^2-1) ) * ( -1*(a^2-1) ) \\ & = ( -1*(a^2-1) )^2 \\ & = ( 1 - a^2 )^2 }

Or, simply, because $a^2 = (-a)^2$...

3. Jan 16, 2016

mattyk

Good point. I'll have to remember that.

4. Jan 16, 2016

Ray Vickson

I don't know why you lost a mark. Both (1-u^2)^2 and (u^2 - 1)^2 are equal to u^4 - 2 u^2 + 1, so IF you performed that expansion you should not have lost a mark.