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Question

# Integrate the rational functions. ∫3x−1(x−1)(x−2)(x−3)dx.

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Solution

## Let 3x−1(x−1)(x−2)(x−3)=A(x−1)+B(x−2)+C(x−3) ⇒3x−1(x−1)(x−2)(x−3)=A(x−2)(x−3)+B(x−1)(x−3)+C(x−1)(x−2)(x−1)(x−2)(x−3)⇒3x−1=A[x2−5x+6]+B[x2−4x+3]+C[x2−3x+2]⇒3x−1=x2(A+B+C)+x(−5A−4B−3C)+(6A+3B+2C) On equating the coefficients of x2, x and constant term on both sides, we get A+B+C =0 ......(i) -5A-4B-3C =3 .......(ii) and 6A+3B+2C=-1 .....(iii) Form Eq.(i), we get A =-(B+C) On putting the value of A in Eqs. (ii) and (iii), we get −5(−(B+C))−4B−3C=3⇒5B+5C−4B−3C=3 ⇒B+2C=3......(iv) and 6(−(B+C))+3B+2C=−1⇒−6B−6C+3B+2C=−1 ⇒−3B−4C=−1..........(v) On solving Eqs. (iv)and (v), we get ⇒C=4 On putting the value of C in Eq. (iv), we get B+2×4=3⇒B=−5 Putting the value of B and C in Eq. (i), we get A +(-5)+4 =0 ⇒A=1∴A=1,B=−5,C=4 Now, ∫3x−1(x−1)(x−2)(x−3)dx=∫(A(x−1)+B(x−2)+C(x−3))dx =∫1(x−1)dx+∫(−5)(x−2)dx+∫4(x−3)dx=log|x−1|−5log|x−2|+4log|x−3|+C(∵1xdx=logx)

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