integral -oo, oo [(2x)/(x^2+1)^2] dx
(a) state why the integral is improper or involves improper integral
*infinite limit of integration
(b) determine whether the integral converges or diverges
converges?
(c) evaluate the integral if it converges
I know f(x)=arctan->f'(x)=1/(1+x^2)
I'm not sure what I do with the 2x in the numerator and the squared demoninator.
(a) The integral is improper because it involves integration over the infinite limits of -∞ to ∞.
(b) To determine whether the integral converges or diverges, we need to check if the integrand approaches a finite value or goes to infinity as x approaches the limits of integration. Let's analyze the integrand:
f(x) = (2x) / (x^2 + 1)^2
As x approaches ∞, the denominator (x^2 + 1)^2 grows much faster than the numerator (2x). Therefore, intuitively, we can expect the integrand to approach zero as x -> ∞. Similarly, as x approaches -∞, the denominator grows faster than the numerator, resulting in the integrand approaching zero.
(c) To evaluate the integral, we can use the substitution technique. The substitution u = x^2 + 1 simplifies the integrand:
du = 2x dx
Rearranging and substituting these values into the integral, we get:
∫ (2x) / (x^2 + 1)^2 dx
= ∫ (1/(x^2 + 1)^2) * (2x dx)
= ∫ (1/u^2) * du
Now, the integral becomes:
∫ (1/u^2) du
Integrating this is straightforward:
∫ (1/u^2) du = -1/u + C
Finally, substituting u back into the result:
= -1/(x^2 + 1) + C
Therefore, the value of the integral is:
∫ (2x) / (x^2 + 1)^2 dx = -1/(x^2 + 1) + C