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Question

Integrate the rational functions.
x(x1)(x2)(x3)dx.

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Solution

x(x1)(x2)(x3)dx
Let x(x1)(x2)(x3)=A(x1)+B(x2)+C(x3)
x(x1)(x2)(x3)=A(x2)(x3)+B(x1)(x3)+C(x1)(x2)(x1)(x2)(x3)x=A[x22x3x+6]+B[x24x+3]+C[x2x2x+2]x=x2(A+B+C)+x(5A4B3C)+(6A+3B+2C)
On comparing the coefficients of x2, x and constant term on both sides, we get
A+B+C=0A=(B+C)........(i)
-5A -4B-3C=1 ...........(ii)
and 6A+3B +2C =0........(iii)
On putting the value of A in Eqs. (ii)and (iii), we get
5((B+C))4B3C=15B+5C4B3C=1B+2C=1......(iv)
and 6((B+C))+3B+2C=03B4C=0........(v)
Multiplying by 3 in Eq. (iv) and then adding in Eq. (v), we get C =32
On putting the value of C in Eq. (iv), we get B+2×32=1B=2.
Now, put the values of B and C in Eq. (i), we get
A2+32=0A=432=12A=12,B=2 and C=32
Now, x(x1)(x2)(x3)dx=A(x1)dx+B(x2)dx+C(x3)dx
=121(x1)dx21(x2)dx+321(x3)dx=12log|x1|2log|x2|+32log|x3|+C


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