Guest

integrate xlogx/(1+x 2 ) 2 under the limits 0 to infinity

integrate   xlogx/(1+x2)2 under the limits 0 to infinity

Grade:12

1 Answers

Badiuddin askIITians.ismu Expert
148 Points
12 years ago

Dear

I = o xlogx/(1+x2)2 dx

first find out

I1 = ∫x/(1+x2)2 dx

     its simple intigration calculate it

I1 = -1/2(1+x2)

now for I =o xlogx/(1+x2)2 dx

take logx as first function  and x/(1+x2)as second function and intigrate by part

I=logx [ -1/2(1+x2)] limit o to ∞      + o 1/x2(1+x2) dx

I=logx [ -1/2(1+x2)] limit o to ∞      + 1/2 o x/x2(1+x2) dx

put x2=t in second part

I=logx [ -1/2(1+x2)] limit o to ∞      + 1/4 [ln t/(t+1)] limit o to

now we have to calculate

I =      Lim x→ [ -logx /2(1+x2)]   - Lim x→ o  [-logx /2(1+x2)] 

                                                         +Lim t→ 1/4 [ln t/(t+1)]  - Lim t→ o 1/4 [ln t/(t+1)]

 third limit is clearly equal to zero

I =      Lim x→ [ -logx /2(1+x2)]   - Lim x→ o  [-logx /2(1+x2)]   - Lim t→ o 1/4 [ln t/(t+1)]

now find out the each limit

Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed  solution very  quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

All the best.
 
Regards,
Askiitians Experts
Badiuddin






Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free