Evaluating Improper Integral of $\frac{x\arctan{x}}{(1+x^2)^2}$

In summary, to evaluate the integral \int^{\infty}_{0}\frac{x\arctan{x}}{(1+x^2)^2}dx, it is suggested to use integration by parts and the substitution u=\arctan x. This leads to an expression of the integral in terms of \int{\frac{dx}{(1+x^2)^2}}, which can then be solved using the substitution x=\tan u.
  • #1
neik
15
0
Evaluate the integral:
[tex]\int^{\infty}_{0}\frac{x\arctan{x}}{(1+x^2)^2}dx[/tex]

can anybody give me some hint? :cry:
Thanks in advance
 
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  • #2
Integrating by parts and substituting [itex]u=\arctan x[/itex] will work.
 
  • #3
I hope u know how to choose the "u" and the "dv' for the part integration...

Daniel.
 
  • #4
I did like this:

Let [tex]u=\arctan{x} \Rightarrow du=\frac{dx}{1+x^2}[/tex]

Let [tex]dv=\frac{x}{(1+x^2)^2}dx \Rightarrow v=-\frac{1}{2(1+x^2)}[/tex]

[tex]\int{udv}=uv-\int{vdu}
=-\frac{\arctan{x}}{2(1+x^2)}dx+\frac{1}{2}\int{\frac{dx}{(1+x^2)^2}}
[/tex]

and then I'm blocked here:
[tex]\int{\frac{dx}{(1+x^2)^2}[/tex]
:cry: :cry: :cry:
 
  • #5
Use the substituion that Galileo prescribed...Denote the second integral by I:

[tex] I=:\int \frac{dx}{(1+x^{2})^{2}} [/tex]

Make the substitution:

[tex] x=\tan u [/tex] and say what u get...

Daniel.
 

FAQ: Evaluating Improper Integral of $\frac{x\arctan{x}}{(1+x^2)^2}$

What is an improper integral?

An improper integral is an integral where at least one of the limits of integration is infinite or the integrand is unbounded at one of the limits of integration. This means that the integral cannot be evaluated using the standard rules of integration and requires special techniques.

What is the integrand in this improper integral?

The integrand in this improper integral is x*arctan(x)/(1+x^2)^2.

How do you evaluate an improper integral?

To evaluate an improper integral, we first identify the type of discontinuity at the limit of integration. Then, we use appropriate techniques such as substitution, integration by parts, or partial fractions to rewrite the integral in a form that can be evaluated using the standard rules of integration. Finally, we take the limit as the variable approaches infinity or negative infinity to get the final value of the integral.

What is the domain of the integrand in this improper integral?

The domain of the integrand is (-∞, ∞) since both x and arctan(x) have no restrictions on their domain and (1+x^2)^2 is always positive.

What are the possible convergence or divergence of this improper integral?

This improper integral may converge or diverge, depending on the value of the integral. If the integral converges, it means that the area under the curve is finite. If the integral diverges, it means that the area under the curve is infinite. The convergence or divergence of this improper integral can be determined by evaluating the limit as the variable approaches infinity or negative infinity.

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