Question

Exercise Resolve into partial fractions x² +1 X 1) (x+1) (x+4) 2​

Answer 1

$\bold{ANSWER≈}$

Let x2+1(x−1)2(x2+2x+2)=A(x−1)2+Bx−1+Cx+Dx2+2x+2

Multiplying by (x−1)2(x2+2x+2)

x2+1=A((x2+2x+2)+B(x−1)(x2+2x+2)+(Cx+D)(x−1)2…….(1)

Comparing coefficients is longer process.

(1) Substitute x=12=5A;A=25

(2) Differentiate w.r.t x and put x=1 ; The term of (x−1)2 and above can be ignored.

2x=25(2x+2)+B{x2+2x+2+(x−1)(2x+2)}⟹putting x=1

2=25∗4+B∗5⟹B=225

(3) Compare coefficient of highest power of x i.e. of x^3

B+C==0⟹C=−225

(4) Comparing constant terms on both sides

1=2A−2B+D⟹D=925

x2+1(x−1)2(x2+2x+2)=25(x−1)2+225(x−1)+9−2x25(x2+2x+2)