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Question

x2(1lnx)ln4xx4dx is equal to

A
12ln(xlnx)14ln(ln2xx2)+C
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B
14ln(ln(x)xlnx+x)12tan1(lnxx)+C
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C
14ln(ln(x)+xlnxx)12tan1(lnxx)+C
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D
14(ln(ln(x)xlnx+x)+tan1(lnxx))+C
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Solution

The correct option is B 14ln(ln(x)xlnx+x)12tan1(lnxx)+C
I=x2(1lnx)ln4xx4dx
=x2(1lnx)x4((lnxx)41)
=1lnxx2((lnxx)41)
Put lnxx=t
(1lnxx2)dx=dt
I=dtt41
=1(t21)(t2+1)dt
=12(t2+1)(t21)(t21)(t2+1)dt
=121t21dt121t2+1dt
I=14ln(t1t+1)12tan1t+C
I=14ln(ln(x)xlnx+x)12tan1(lnxx)+C

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