Задача 37254 ...

yelymcheav

2019-05-19 12:09:24

Найти интеграл ∫ ((x+1)dx) / (x(x^2+2x+2))

sova

2019-05-19 12:26:56

Раскладываем дробь на простейшие:

(x+1)/(x*(x^2+2x+2))= (A/x)+(Mx+N)/(x^2+2x+2)

x+1= A*(x^2+2x+2)+(Mx+N)*x

0=A+M
1=2A+N
1=2A

A=1/2
M=-1/2
N=0


= (1/2)∫ dx/(x+1) - (1/2) ∫ xdx/(x^2+2x+2)=

=(1/2)ln|x+1| - (1/4) ∫( 2x+2-2)dx/(x^2+2x+2)=

=(1/2)ln|x+1| - (1/4)ln|x^2+2x+2| +(1/2) ∫ dx/((x+1)^2+1)=

= [b](1/2)ln|x+1| - (1/4)ln|x^2+2x+2| +(1/2) arctgx + C[/b]