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Question

Integrate the rational functions.
5x(x+1)(x24)dx.

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Solution

5x(x+1)(x24)dx=5x(x+1)(x+2)(x2)dx
Let 5x(x+1)(x24)=A(x+1)+B(x+2)+C(x2)......(i)
=A(x+2)(x2)+B(x+1)(x2)+C(x+1)(x+2)(x+1)(x+2)(x2)5x=A(x24)+B(x2+x2x2)+C(x2+x+2x+2)5x=x2(A+B+C)+x(B+3C)+(4A2B+2C)
On comparing the coefficients of x2,x and constant term on both sides, we get

A+B+C =0 ....(ii)
-B +3C =5....(iii)
and -4A-2B +2C =0 ....(iv)
Multiply by 4 in Eq. (ii) and then adding with Eq. (iv), we get
2(B+3C)=0....(v)
On adding Eqs. (iii)and (v), we get C=56
On putting the value of C in Eq. (v), we get B=52
On putting the values of B and C in Eq. (ii), we get A=53
5x(x+1)(x2)(x+2)dx=53(x+1)dx52(x+2)dx+56(x2)dx=531(x+1)dx521(x+2)dx+561(x2)dx=53log|x+1|52log|x+2|+56log|x2|+C


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