1. ∫xn = xn+1/n+1 20.lim sinx/x =1
2. ∫1/x =ln x+c 21.tgx =sinx/cosx
3. ∫ax dx = ax /ln a 22.sin(-x)=-sinx; cos(-x)=cosx
4. ∫ ex = ex 23.sin2x=2sinx.cosx
5. ∫cosx =sinx 24.cos2x=cos2x-sin2x
6. ∫1/cos2x =tgx 25. (logax)‘=1/x.log a
7. ∫1/√1- x2= arcsinx+c 26. sin2x=1-cos2x/2
8. ∫1/ x2 +1= arctgx 27.cos2x=1-cos2x/2
9. ∫ u.v´= u.v- ∫u´.v 28.Eul.s. a>0:
10.∫f´x/fx= ln|fx|+c t= √ax2+bx+c + x√a ;c>0
11.∫f´x/√fx= 2.√fx+c xt= √ax2+bx+c+√c
12.∫1/ x2+a2 =1/a.arctgx/a+c
19.∫1/ √x2+a2 =ln|x+√x2+a2|
13.∫ 1/ √a2-x2 = arcsinx/a
14.∫√a2-x2 = a2 /2(arcsin x/a +x/a √1- x2/a2 )
15.∫A/(x-a) k = a.) ak k=1 => A ln|x-a|+c
ak k>1 => A ∫1/(x-a) k =|subst: t= x-a;x=a+t;dx=dt|=
=A (t –k+1 /-k+1)=A/(k-1).(x-a) k-1
16. ∫1/ x2 +px+q = ∫ 1/(x+p/2) 2+q-p2/4
| subst: x+p/2 =t; q-p2/4=a|
=1/ t2+a2 =tvar 12
17. ∫ Mx+N/x2 +px+q = M/2 ∫2x+p/x2+px+q+(N-p.M/2).
∫ 1/x2 +px+q =
M/2 ln| x2 +px+q| + (N-p.M/2).∫ 1/(x+p/2) 2+q-p2/4
18. ∫ x/(x2+a2) k = ak k=1 => ½∫ 2x/(x2+a2) =½ln|x2+a2|
ak k>1 tak subst: t = x2+a2 ;dx=dt/2; => ½∫ 1/tk dt=
½ t –k+1/-k+1
19. ∫1/ √x2 +px+q = ∫1/√ t2+a2 ...... 20. ∫1/1- x2= 1/2 ln|1+x/1-x|
29. ∫ x2/(x2+a2) k=perpart. U=x;v= x/(x2+a2) k=>∫v=-1/2(k-1) /
(x2+a2) k-1.....
30. ∫√(x+bx+c)= {(Ax+Bx+C).√f(x+bx+c)}´+k ∫1/ x+bx+c=
zderivujeme a integral sa odstr. Potom cely vyraz /.√
31. ∫sin mx.cos nx =∫1/2sin(m-n)+sin(m+n)
∫cos mx.cos nx =∫1/2cos(m-n)+cos(m+n)
∫sin mx.sin nx =∫1/2cos(m-n)-cos(m+n)
L.O.Pr:1.limf= limg =0
Tak Ex. lim f´/lim q´
2.limg= nekon. Tak Ex.
lim f´/lim g´
a. limf=limg=nekon a mam limf-g tak
na podiel 1/g-1/f
1/f.g
b. limf=0,limg=nek. a mam f.g => f/ 1/g
c. fg=eg.lnf |Fermatova:Ak f nadobuda v
bode a max(min)hodn. a ma v tom bode deriv.tak f’a=0
Lopitalkami urcujeme taketo tvary:
1.limf/g ak limf=0 al.