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integrate {[(x 2 +1) 1/2 [ln(x 2 +1) - 2lnx]]/x 4 } w.r.t x

integrate  {[(x2+1)1/2[ln(x2+1) - 2lnx]]/x4} w.r.t x

Grade:12th Pass

2 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Suchita,


Take x2 from the root, and combine the two log, you will have {(1+1/x2)1/2*ln(1+1/x2)}/x3 dx.

Now make the substitution 1+1/x^2 = t, and you have dx/x^3 = -3dt.

And the integral will be ∫(-3)*t*lnt dt, which is directly integrated by parts

= -3/2 { t*2 lnt - ∫tdt }. And that solves the problem, substitute back for t = 1 + 1/x^2.


Hope that helps,


All the best,


Ashwin (IIT Madras).

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Suchita,


Please note, here again the method of integration is the same as mentioned above.


Where 2lnx = lnx2

And ln(1+x2) - lnx2 = ln[(1+x2)/x2] = ln(1+1/x2)


And when x^2, is taken from the square root term, it cancels one x in the Dr, leaving x3.


So sub 1+1/x2 = t. Which will give dx/x3 = -dt/2


Please be careful with such calculations (previously there was a mistake while doing it very fast, and hence a mistake in the constant term. Please do take time while practicing questions to not make such very silly mistake).


Hope that helps.


All the best.


Ashwin (IIT Madras).

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