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Question

x2(1lnx)(lnx)4x4 dx for x>0 equals
(C is an integration constant)

A
12lnxlnx14ln(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=t1lnxx2=dt
I=1t41 dt

=dt(t2+1)(t21)

=12(t2+1)(t21)(t2+1)(t21) dt

=12(dtt21dtt2+1)

=12(12lnt1t+1tan1t)+C

=14lnlnxxlnx+x12tan1(lnxx)+C

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