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Question

The value of the integral x(x1)(x2+4)dx (where C is integration constant)

A
15log|x1|110log|x2+4|+25tan1x2+C
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B
15log|x1|110log|x2+4|+45tan1x2+C
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C
15log|x1|+110log|x2+4|+25tan1x2+C
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D
15log|x+1|110log|x2+4|+25tan1x2+C
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Solution

The correct option is A 15log|x1|110log|x2+4|+25tan1x2+C
Given x(x1)(x2+4)dx

Let x(x1)(x2+4)=Ax1+Bx+Cx2+4(1)
x=A(x2+4)+(Bx+C)(x1)(2)
Putting x=1 in equation (2), we get 1=5A
Putting x=0 in equation (2), we get 0=4AC
Putting x=1 in equation (2), we get 1=5A+2B2C

Solving these equations, we obtain A=15,B=15 and C=45
Substituting the values of A,B and C in equation (1), we obtain
x(x1)(x2+4)=15(x1)+15x+45x2+4
=15(x1)15(x4)(x2+4)
I=151x1dx15x4x2+4dx
I=151x1dx1102xx2+4dx+451x2+4dx
I=15log|x1|110log|x2+4|+45×12tan1x2+C
I=15log|x1|110log|x2+4|+25tan1x2+C

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