A clarification on a step in an integration question

[mattyk]

Homework Statement

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

Homework 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.

The Attempt at a Solution

Is this possible and if so how?

 

[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 ##...

 

[mattyk]

Good point. I'll have to remember that.

 

[Ray Vickson]

Homework Statement

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

Homework 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.

The Attempt at a Solution

Is this possible and if so how?

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.