Can a Shift Simplify the Triple Integral of cos(u+v+w)?

[Nah_Roots]

\int \int \int cos(u + v + w)dudvdw (all integrals go from 0 to pi).

I've tried using u substitution for each integral but I end up with a huge integral.

 

[HallsofIvy]

I don't see why. To integrate cos(u+v+w)du, let x= u+ v+ w so dx= du. The integral becomes $\int$ cos(x)dx= -sin(x)+ C= -sin(u+ v+ w)+ C. Evaluating that at 0 and pi gives -sin(pi+ v+ w)+ sin(v+ w). But sin(x+ pi)= -sin(x) so that is just 2 sin(v+w).

Now integrate 2sin(v+w) dv by letting x= v+ w so dx=dv.

 

[Nah_Roots]

I don't understand sin(x+ pi)= -sin(x). Is that an identity I am forgetting about?

 

[HallsofIvy]

Do you know what the graph of y= sin(x) looks like?

 

[Nah_Roots]

Oh, I see. It's a shift, correct?