How do you write the partial fraction decomposition of the rational expression #(x^2 + 1)/(x(x + 1)(x^2 + 2))#?

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Answer 1 · 2015-12-14T04:10:04

#1/(2x)-2/(3(x+1))+(x+2)/(6(x^2+2))#

#(x^2 + 1)/(x(x + 1)(x^2 + 2))=A/x+B/(x+1)+(Cx+D)/(x^2+2)#

#x^2+1=A(x+1)(x^2+2)+B(x)(x^2+2)+(Cx+D)(x)(x+1)#

#x^2+1=Ax^3+Ax^2+2Ax+2A+Bx^3+2Bx+Cx^3+Cx^2+Dx^2+Dx#

#x^2+1=x^3(A+B+C)+x^2(A+C+D)+x(2A+2B+D)+1(2A)#

Thus, #{(A+B+C=0),(A+C+D=1),(2A+2B+D=0),(2A=1):}#

Solve this to see that #{(A=1/2),(B=-2/3),(C=1/6),(D=1/3):}#

Therefore,

#(x^2 + 1)/(x(x + 1)(x^2 + 2))=1/(2x)-2/(3(x+1))+(x+2)/(6(x^2+2))#