integrali

Neodređeni integral

sintaksa naredbe

(%i1)

integrate(f(x),x);

(%o1) (%o1) integrate(f(x),x)

neki tablični integrali

(%i2)

integrate(x^n,x);

(%i3)

integrate(1/x,x);

(%i4)

integrate(a^x,x);

(%i5)

integrate(1/(a*x+b),x);

(%i6)

integrate(1/(x^2+a^2),x);

(%i7)

integrate(1/(x^2-a^2),x);

(%i8)

logcontract(%);

(%i9)

integrate(diff(f(x),x)/f(x),x);

primjer 1

(%i10)

integrate(1/x^(3/4),x);

primjer 2

(%i11)

integrate((x-3)^2/x^5,x);

(%i12)

expand(%);

primjer 3

(%i13)

integrate(5*%e^x-3*sin(x),x);

primjer 4

(%i14)

integrate(3^x*%e^x,x);

(%o14) 3^((1+1/log(3))*x)/((1/log(3)+1)*log(3))

(%i15)

expand(%);

primjer 5

(%i16)

integrate((3-2*x)^8,x);

primjer 6

(%i17)

integrate((x-2)^(3/4),x);

primjer 7

(%i18)

integrate(x*7^(x^2),x);

primjer 8

(%i19)

integrate(x/sqrt(1-x^2),x);

primjer 9

(%i20)

integrate(x^2/(1+x^6),x);

primjer 10

(%i21)

integrate((1+log(x))^(1/3)/x,x);

primjer 11

(%i22)

integrate((1-sin(x))/(x+cos(x)),x);

primjer 12

(%i23)

integrate(1/(%e^x+1),x);

primjer 13

(%i24)

integrate(log(x),x);

primjer 14

(%i25)

integrate(x*cos(3*x),x);

primjer 15

(%i26)

integrate(x^2*%e^(3*x),x);

primjer 16

(%i27)

integrate(x^5*log(x),x);

primjer 17

(%i28)

integrate(%e^x*cos(x),x);

primjer 18

(%i29)

integrate(1/(3*x^2+5),x);

primjer 19

(%i30)

integrate(1/(3-2*x),x);

primjer 20

(%i31)

integrate(1/(x^2-3),x);

(%o31) log((2*x-2*sqrt(3))/(2*x+2*sqrt(3)))/(2*sqrt(3))

primjer 21

(%i32)

integrate((1-3*x)/(3+2*x),x);

primjer 22

(%i33)

integrate(1/(3*x^2+x-4),x);

primjer 23

(%i34)

integrate(1/(3*x^2+x+4),x);

(%o34) (2*atan((1+6*x)/sqrt(47)))/sqrt(47)

primjer 24

(%i35)

integrate((x^2+5*x-4)/(5*x+3),x);

(%o35) (44*x+5*x^2)/50-(166*log(5*x+3))/125

primjer 25

(%i36)

integrate((5*x+3)/(x^2+5*x-4),x);

(%o36) (5*log(x^2+5*x-4))/2-(19*log((5-sqrt(41)+2*x)/(2*x+sqrt(41)+5)))/(2*sqrt(41))

Određeni integral

(%i37)

load(draw)$

(%i38)

set_draw_defaults(
   grid=true,
   xaxis=true,
   yaxis=true,
   xaxis_width=2,
   yaxis_width=2,
   xaxis_type=solid,
   yaxis_type=solid)$

sintaksa naredbe

(%i39)

integrate(f(x),x,a,b);

(%o39) (%o39) integrate(f(x),x,a,b)

********************************
** primjer 1 - sinus funkcija **
********************************

(%i40)

wxdraw2d(fill_color = orange,xrange=[-1,7],yrange=[-1.3,1.3],
      filled_func = 0,
      explicit(sin(x),x,0,2*%pi),
      filled_func = false, line_width=2,
      explicit(sin(x),x,0,2*%pi));

određeni integral funkcije sinus na [0,2*pi] je jednak nula

(%i41)

integrate(sin(x),x,0,2*%pi);

površina

(%i42)

p1:integrate(sin(x),x,0,%pi);

(%i43)

p2:-integrate(sin(x),x,%pi,2*%pi);

ukupna površina

(%i44)

p1+p2;

**********************************************************
** primjer 2 - površina ispod grafa prirodnog logaritma **
**********************************************************

(%i45)

wxdraw2d(fill_color = orange,xrange=[-1,7],yrange=[-3,3],
      filled_func = 0,
      explicit(log(x),x,2,5),
      filled_func = false, line_width=2,
      explicit(log(x),x,0,7));

ukupna površina

(%i46)

integrate(log(x),x,2,5);

(%i47)

%,numer;

******************************************************************************
** primjer 3 - površina omeđena pravcima y=x i x=3, krivuljom y=1/x i x-osi **
******************************************************************************

(%i48)

wxdraw2d(fill_color = orange,xrange=[-1,4],yrange=[-1.5,4],
      filled_func = 0,
      explicit(x,x,0,1), explicit(1/x,x,1,3),
      filled_func = false, line_width=2,
      explicit(x,x,-1,4), explicit(1/x,x,0.1,4));

površina

(%i49)

p1:integrate(x,x,0,1);

(%i50)

p2:integrate(1/x,x,1,3);

ukupna površina

(%i51)

p1+p2;

************************************************************************
** primjer 4 - površina omeđena parabolom f(x)=x^2 i pravcem g(x)=x+2 **
************************************************************************

(%i53)

f(x):=x^2;
g(x):=x+2;

(%i54)

wxdraw2d(fill_color = orange,xrange=[-3,4],yrange=[-1.5,5],
      filled_func = x^2,
      explicit(x+2,x,-1,2),
      filled_func = false, line_width=2,
      explicit(x^2,x,-2.5,2.5), explicit(x+2,x,-2.5,2.7));

presjek grafova

(%i55)

solve(f(x)=g(x),x);

ukupna površina

(%i56)

integrate(g(x)-f(x),x,-1,2);

**********************************************************************************
** primjer 5 - površina omeđena krivuljama y=1/x, y=x^2 i y=4 u prvom kvadrantu **
**********************************************************************************

(%i57)

wxdraw2d(fill_color = orange,xrange=[-3,4],yrange=[-1.5,5],
      filled_func = 4, explicit(1/x,x,1/4,1),
      filled_func = 4, explicit(x^2,x,1,2),
      filled_func = false, line_width=2,
      explicit(x^2,x,-2.5,2.5), explicit(1/x,x,0.1,5),explicit(4,x,-2.5,3.5));

površina

(%i58)

p1:integrate(4-1/x,x,1/4,1);

(%i59)

p2:integrate(4-x^2,x,1,2);

ukupna površina

(%i60)

p1+p2;

(%i61)

%,numer;

Elastičnost i granični troškovi

***************
** PRIMJER 1 **
**************************************************************************
Odredite funkciju potražnju q(p) ako je njezina elastičnost jednaka E=-2p
i vrijedi q(0)=2.
**************************************************************************

sve funkcije q(p) čija je elastičnost jednaka -2p

(%i62)

rj:ode2(p/q*'diff(q,p)=-2*p,q,p);

funkcija za koju još vrijedi i q(0)=2

(%i63)

ic1(rj,p=0,q=2);

***************
** PRIMJER 2 **
**************************************************************************
Zadana je funkcija graničnih troškova Tg(Q)=(1+Q)e^(-Q). Odredite funkciju
troškova ako fiksni troškovi iznose 100.
**************************************************************************

(%i64)

tr:integrate((1+Q)*%e^(-Q),Q);

(%i65)

solve(subst(0,Q,tr)+C=100,C);

funkcija troškova

(%i66)

tr+102;

Created with wxMaxima.