Question

Hi, I'm having some trouble understanding the steps in the solution to an integration problem involving manipulating a coefficient. See the screenshot below. I think it's just algebra but for some reason I can't follow along. Any help would be appreciated.

Answer 1

How to progress from the first line to the second?

Answer 2

I think the definite integral should give \[I=\int\limits_{0}^{1} \cos(\pi x)dx[-\frac{ 1 }{ \pi }(\cos(\pi-\pi x)-(-\frac{ 1 }{ \pi } )]\]

but wouldn't that result in this:
\[I=\int\limits_{0}^{1} (-\frac{ 1 }{ \pi })cos(\pi x)dx(\cos(\pi-\pi x)-1)\]
instead of what has been shown in line 2 above?

Answer 3

there is a - *minus* missing there,on 2nd line, you are correct.

Answer 4

Thank you @hartnn , glad it's not as confusing as I thought it was. There is one more thing I am not sure about though - how the second line progresses to the third. How does cos(πx)dx(cos(π−πx)−1) become (((cos(πx))^2)-(cos(πx))dx)? More specifically it is the (cosπx)*(cos(π-πx))=((cos(πx))^2) that I am confused about. Is there some trig identity I have forgotten?

Answer 5

Ah, nevermind, I did forget that cos(π-πx)=cos(π)cos(πx)-sin(π)sin(πx) . Thanks again for your help!

Answer 6

or \(\cos (\pi -\theta)=-\cos \theta\)