(a) How do we express a rational function r as a partial frac- tion decomposition? (b) Give the form of the partial fraction decomposition. (i)2x(x−5)(x−1)2 (ii)2x(x−5)(x2+1) (iii)3x+1x(x2+1)2

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(a) How do we express a rational function r as a partial frac- tion decomposition?   (b) Give the form of the partial fraction decomposition.   (i)2x(x−5)(x−1)2    (ii)2x(x−5)(x2+1)    (iii)3x+1x(x2+1)2

Brittany Patton · Accepted Answer

Step 1   (a)   A rational function is a function of the form  r(x)=P(x)Q(x)  Where P(x) and Q(x) are polynomials in x with  Q(x)≠0  Rational fractions can be integrated by splitting into partial fractions.   Step 2   Consider the partial fraction,  2x(x−5)(x−1)2  To write the given fraction in partial fraction decomposition,  2x(x−5)(x−1)2=Ax−5+Bx−1+C(x−1)2  Where  A,  B   and  C   are  constant.  Consider the partial fraction,  2x(x−5)(x2+1)  To write the given fraction in partial fraction decomposition,  2x(x−5)(x2+1)=Ax−5+Bx+C(x2+1)  Where  A,  B   and  C   are  constant.  Step 4  Consider the partial fraction,  3x+1x(x2+1)2  To write the given fraction in partial fraction decomposition,  3x+1x(x2+1)2=Ax+Bx+C(x2+1)+Dx+E(x2+1)2  Where  A,  B   and  C   are  constant.

(a) How do we express a rational function r as a partial frac- tion decompositio

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