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Question

Evaluate ∫x dx(x−1)(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(x−1|−110log(x2+4)+25tan−1(x2)+CWe can write the integrand as: let, x(x−1)(x2+4)=Ax−1+Bx+cx2+4Solving this, we getA=15, B=−15 and C=45. Now, substituting the values of A, B and C,x(x−1)(x2+4)=15(x−1) +−15x+45x2+4 =15(x−1)−15(x−4)(x2+4) Or, we can write I=15∫dxx−1−15∫x−4x2+4dx =15∫dxx−1−110∫2xx2+4dx +45∫1x2+4dx =15log(x−1|−110log(x2+4∣∣ +45×12tan−1(x2)+C=15log(x−1|−110log(x2 + 4) +25tan−1(x2)+C (∵(x2+4∣∣ = x2 + 4)

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