"arctant" - 1 õppematerjal

Matemaatiline analüüs 1 kordaisküsimuste vastused

I=? X2+px+q=täiruudu er (x2+2x p/2 +p2/4)- p2/4 +q =(x+p/2)2 +(q-p2/4)= (q-p2/4) p2 p2 p p2 *[(x+p/2 / q - )2 +1]; I=1/ q - arctan x+ / q - +C 4 4 2 4 31.F-nide R(sinx, cosx) int-misest Def R(sinx, cosx)=> y=sinx; z=cosx=>tanx=y/z, cotx=z/y=>R(y,z)-rats y,z suhtes. *Nt sinx/cosx+1 =y/z+1; *I universaalne asendus R (sin x; cos x)dx = asendus:t=tanx/2, x/2=arctant, x=2 arctant, dx= 2/1+t2 *dt, sinx=2t/1+t2, cosx=1-t2/1+t2; = R (2t / 1 + t ;1 - t / 1 + t )2 / 1 + t * dt II 2 2 2 2 dt R(tan x)dx = R(t ) 1 + t 2 III R(sin x) cos xdx = R(t )dt , as t=sinx, dt =sinx'dx=cosxdx(saab ka vastupidi, st sulgudes on cosx) IV sin mxcosnxdx: a)m=2k+1>0=> sin x cos x sin xdx = -(1 -t )t dt , as t=cosx, dt =-sinxdx, 2k n 2 n