Find the partial fraction decomposition. \frac{x^{2}+x-6}{(x^{2}+1)(x-1)}

sjeikdom0

sjeikdom0

Answered question

2021-09-26

Find the partial fraction decomposition.
x2+x6(x2+1)(x1)

Answer & Explanation

komunidadO

komunidadO

Skilled2021-09-27Added 86 answers

Step 1
Formations of Partial fractions is method to separate a fraction into its components such that when the components are added together, they give back the original fraction.
This method is very useful in integration or while performing Fourier transforms, Laplace transforms of various functions .
Step 2
The given function is represented as x2+x6(x2+1)(x1).
Partial fractions for the above function would be of the form Ax+Bx2+1+Cx1
Equating two functions to find the value of A, B and C.
x2+x6(x2+1)(x1)=Ax+Bx2+1+Cx1
=(Ax+B)(x1)+C(x2+1)(x2+1)(x1)
x2+x6=(A+C)x2+(A+B)x+(B+C)
Equating coefficients:
A+C=1
A+B=1
B+C=6
B+C=2
2C=4
C=2
A=3
B=4
Hence the partial fraction formed is: x2+x6(x2+1)(x1)=3x+4x2+12x1

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