How do you test the improper integral #int (x(1+x^2)^-2)dx# from #[0,oo)# and evaluate if possible?

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Answer 1 · 2017-03-17T22:50:34

#int_0^oo x(1+x^2)^-2dx = 1/2#

Evaluate first the indefinite integral:

#F(x) = int x(1+x^2)^-2dx#

by substituting #(1+x^2) = t#, so that #dt = 2xdx#:

#int x(1+x^2)^-2dx = 1/2 int t^-2dt =-1/(2t)+C =-1/(2(1+x^2))+C#

Now we have:

#F(0) = -1/2+C#

#lim_(x->oo) F(x) = C#

So the indefinite integral is convergent and we have:

#int_0^oo x(1+x^2)^-2dx = (lim_(x->oo) F(x)) - F(0) = 1/2#