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Question

Evaluate x dx(x1)(x2+4)

A
15log(x1|+110log(x2+4)+25tan1(x2)+C
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B
15log(x1|110log(x2+4)+25tan1(x2)+C
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C
15log(x1|110log(x2+4)+25tan1(x2)+C
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D
15log(x1|110log(x2+4)25tan1(x2)+C
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Solution

The correct option is B 15log(x1|110log(x2+4)+25tan1(x2)+C
We can write the integrand as:
let, x(x1)(x2+4)=Ax1+Bx+cx2+4Solving this, we getA=15, B=15 and C=45.
Now, substituting the values of A, B and C,x(x1)(x2+4)=15(x1) +15x+45x2+4 =15(x1)15(x4)(x2+4)
Or, we can write
I=15dxx115x4x2+4dx =15dxx11102xx2+4dx +451x2+4dx =15log(x1|110log(x2+4 +45×12tan1(x2)+C=15log(x1|110log(x2 + 4) +25tan1(x2)+C ((x2+4 = x2 + 4)

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