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Question

For x>0 and arbitrary constant of integration C, x2(1lnx)(lnx)4x4dx equals

A
14lnxlnx14ln(lnx)2x2+C
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B
14lnlnxxlnx+x12tan1(lnxx)+C
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C
14lnlnx+xlnxx+12tan1(lnxx)+C
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D
14(lnlnxxlnx+x+tan1(lnxx))+C
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Solution

The correct option is B 14lnlnxxlnx+x12tan1(lnxx)+C
I=x2(1lnx)x4((lnxx)41)dx=1lnxx2((lnxx)41)dx

Put lnxx=t1lnxx2dx=dt

I=dtt41=dt(t2+1)(t21)=12(t2+1)(t21)(t2+1)(t21)dtI=12(dtt21dtt2+1)=12(12lnt1t+1tan1t)+C=14lnlnxxlnx+x12tan1(lnxx)+C

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