# How do you evaluate the integral #int1/(1+x^2)^2 dx# from #-oo# to #oo#?

First, for simplicity's sake, examining without the bounds:

The first integral can be solved either through another substitution or by inspection.

Reapplying the bounds:

Taking the limits at both infinities:

By signing up, you agree to our Terms of Service and Privacy Policy

To evaluate the integral ∫1/(1+x^2)^2 dx from -∞ to ∞, we can use the method of complex analysis. We first define a contour integral by integrating the function 1/(1+z^2)^2 over a semicircular contour in the upper half-plane, closing the contour with a straight line along the real axis from -R to R, and letting the radius R approach infinity. By applying Cauchy's residue theorem, we find that the integral along the semicircular contour tends to zero as R approaches infinity. Thus, the integral over the entire contour is equal to the sum of the residues of the function within the contour. The only singularity enclosed by the contour is at z=i, where the residue can be calculated using the formula Res(f(z), i) = lim(z→i) [(z-i)f(z)]. After evaluating the residue, the integral can be expressed as 2πi times the residue, resulting in the final answer.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the integral of #int x / (x^2-9) dx# from 1 to infinity?
- How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function #f(x)=int sqrt(4+sec(t))dt# from #[x, pi]#?
- How do you evaluate the integral #int 1/x dx# from 0 to 1 if it converges?
- What is the antiderivative of #ln(x)^2#?
- If #a_k in RR^+# and #s = sum_(k=1)^na_k#. Prove that for any #n > 1# we have #prod_(k=1)^n(1+a_k) < sum_(k=0)^n s^k/(k!)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7