∫3x−1(x−1)(x−2)(x−3)dx is equal to

3x1(x1)(x2)(x3)dx is equal to

  1. A

    log|x1|5|log|x2|+4log|x3+C

  2. B

    |log|x1|log|x2|+4|log|x3|+C

  3. C

    5log|x1|log|x2|+4log|x3|+C

  4. D

    None of the above

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    Solution:

    Let 3x1(x1)(x2)(x3)=A(x1)+B(x2)+C(x3)
    3x1(x1)(x2)(x3)=A(x2)(x3)+B(x1)(x3)+C(x1)(x2)(x1)(x2)(x3)
    3x1=Ax25x+6+Bx24x+3+Cx23x+23x1=x2(A+B+C)+x(5A4B3C)+(6A+3B+2C)
    On equating the coefficients of x2 ,x and constant term on both sides, we get
                A+B+C=0                     .....(i)          5A4B3C=3            .....(ii) and     6A+3B+2C=1         .....(iii)
    From Eq. (i), we get A = -(B + C)
    On putting the value of A in Eqs. (ii) and (iii), we get
         5{(B+C)}4B3C=3
        5B+5C4B3C=3    6+2C=3                            .....(iv) and     6{(B+C)}+3B+2C=1    6B6C+3B+2C=1    3B4C=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
    On putting the value of Band C in Eq. (i), we get
    A+(5)+4=0A=1     A=1,B=5,C=4
    Now, 
    3x1(x1)(x2)(x3)dx=A(x1)+B(x2)+C(x3)dx    =1(x1)dx+(5)(x2)dx+4(x3)dx=log|x1|5log|x2|+4|log|x3+C                                        1xdx=logx
        

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