Let Un=∫ xcos^n x dx from o to pi/2
Un=∫ xcos^n x dx=int.xcos^(n-1) x cos x dx
let u=cos^(n-1) x and dv\dx=xcosx
du\dx=(n-1)cos^(n-2)x(-sin x) and v=int. xcosx
du\dx=-(n-1)sinxcos^(n-2)x and v=xsinx+cosx
Un=[(xsinx+cosx)cos^(n-1)x] 0 to pi\2 +(n-1)int. (xsinx+cosx)sinxcos^(n-2)x dx
=-1+(n-1)int. xsin²xcos^(n-2)x dx+(n-1)int. sinxcos^(n-1)x dx
=-1+(n-1)int. xcos^(n-2)x dx-(n-1)int. xcos^n x dx+(n-1)int. sinxcos^(n-1) x dx
=-1+(n-1)U_(n-2)-(n-1)Un+(n-1)int. sinxcos^(n-1) x dx
Un(1+(n-1))=-1+(n-1)U_(n-2)+(n-1)int. sinxcos^(n-1) x dx
Un(n)=-1+(n-1)U_(n-2)+(n-1)int. sinxcos^(n-1) x dx
consider I=int. sinxcos^(n-1) x dx
let u=cos^(n-1)x dx and dv\dx=sinx
du\dx=(n-1)cos^(n-2)x(-sinx)=-sinx(n-1)cos^(n-2)x and v=-cosx
I=[-cos^n x] 0 to pi\2 -(n-1)int. sinxcos^(n-1) x dx
I(1+(n-1) )=1
I=1\n
back to:
Un(n)=-1+(n-1)U_(n-2)+(n-1)int. sinxcos^(n-1) x dx
Un(n)=-1+(n-1)U_(n-2)+(n-1)(1\n)
Un(n)=-1+(n-1)U_(n-2)+1-1\n
Un(n)=(n-1)U_(n-2)-1\n
Un=(n-1)\n U_(n-2)-1\n²
a bit messy, might of made error some where but looks rite lol
