Question

integral de x²-3x+5/x³-5x²    dx

Answer 1

x/(x+1)(x+3)(x+5) = 
A/(x+1)+B/(x+3)+C/(x+5) = 
A(x+3)(x+5)+B(x+1)(x+5)+C(x+1)(x+3)=x 
A(x²+3x+5x+15)+B(x²+x+5x+5)+C(x²+x+3x+... 
A(x²+8x+15)+B(x²+6x+5)+C(x²+4x+3)=x 
x²(A+B+C)+x(8A+6B+4C)+15A+5B+3C=x 

A+B+C=0 
8A+6B+4C=1 
15A+5B+3C=0 

A+B+C=0 
-4A-4B-4C=0 
8A+6B+4C=1 
----------------------- 
4A+2B=1 

A+B+C=0 
-3A-3B-3C=0 
15A+5B+3C=0 
----------------------- 
12A+2B=0 

12A+2B=0 - 
4A+2B=1 
-------------------- 
8A=-1 
A=-1/8 

4A+2B=1 
4.(-1/8)+2B=1 
-1/4+2B=1 
2B=1+1/4 
2B=5/4 
B=5/8 

A+B+C=0 
-1/8+5/8+C=0 
4/8+C=0 
C=-1/2 

Voltando,temos: 
x/(x+1)(x+3)(x+5) = 
A/(x+1)+B/(x+3)+C/(x+5) = 
-1/8(x+1)+5/8(x+3)-1/2(x+5) = 
⌠xdx/(x+1)(x+3)(x+5) = 
-1/8⌠dx/(x+1)+5/8⌠dx/(x+3)-1/2⌠dx/(x+5... 
-1/8ln|x+1|+5/8ln|x+3|-1/2ln|x+5|+c= 
1/8[5ln|x+3|-ln|x+1|]-1/2ln|x+5|+c= 
1/8[5ln|x+3|-ln|x+1|-4ln|x+5|]+c= 
1/8[ln|x+3|^5-ln|x+1|-ln|x+5|^4]+c= 
1/8[ln(|x+3|^5/|x+1||x+5|^4)]+c