Ratsional funksiyani (kasrni) integrallash. Ba`zi bir trigonometrik ifodalarni integrallash

> m:=3:value(Ik2);
> m:=4:Ik2:=value(Ik2);

> m:=4:a:=1:Ik2:=value(Ik2);

n=2,3 bo`lganda integralini formula asosida topish.


1) > restart;
> Ik3:=int((A*(t-p/2)+B)/(t^2+a^2)^m,t);

> m:=2:Ik3:=value(Ik3);

> m:=3:Ik3:=value(Ik3);

2) > A:=4:B:=5:p:=6:q:=25:a:=sqrt(4*q-p^2)/2:
> with(student):
> changevar(t=x+p/2, (Ik3, t), x);

3) n=2,3 bo`lganda bevosita topish.


> restart;
> Ik4:=int((4*x+5)/(x^2+6*x+25)^2,x);

> Ik5:=int((4*x+5)/(x^2+6*x+25)^3,x);

3-misol. integralni toping.
Yechish:





hosil bolgan tenglikning chap va o`ng ko`phadlarining darajalari bo`icha mos koeffitsientlarni tenglashtirib quyidagi sistemani tuzamiz:

Sodda kasirlarning koeffitsientlarni topish va integrallash:
> restart;
> f:=x->(1-x^3)/(x^5+x^2);
> y1:=x->A1/x+A2/x^2+A3/(x+1)+(A4*x+A5)/(x^2-x+1);

> p:=simplify(y1(x));

> pol1:=f(x)*(x^5+x^2); pol2:=p*x^2*(x+1)*(x^2-x+1);


> kp0:=coeff(pol1,x,0)=coeff(pol2,x,0);
> kp1:=coeff(pol1,x,1)=coeff(pol2,x,1);
> kp2:=coeff(pol1,x,2)=coeff(pol2,x,2);
> kp3:=coeff(pol1,x,3)=coeff(pol2,x,3);

> kp4:=coeff(pol1,x,4)=coeff(pol2,x,4);

> k:=solve({kp0,kp1,kp2,kp3,kp4},{A1,A2,A3,A4,A5});

> A1:=0; A4:=-2/3; A3:=2/3; A5:=-2/3; A2:=1;y1(x);


> int(y1(x),x);

Topilgan koeffitsientlar asosida berilgan kasirni sodda kasirlarga ajratilgan ko`rinishin yozamiz:



Bebosita sodda kasirlarga ajratish: