1. integral -oo, oo [(2x)/(x^2+1)^2] dx
2. integral 0, pi/2 cot(theta) d(theta)
(a) state why the integral is improper or involves improper integral
(b) determine whether the integral converges or diverges
converges?
(c) evaluate the integral if it converges
CONFUSE: how would I know the integral converges or diverges without
evaluating the integral? I am not sure if I evaluated the integral correctly.
(a) infinite limit of integration
(b) diverges?
lim -1/(x^2+1) -oo, b = oo?
b->oo
(a) integrand has an infinity discontinuity at x=0
(b) diverges?
lim cos(theta)/sin(theta) d(theta)
b->oo
= lim ln(sin(theta)) d(theta) b, pi/2
b->oo
=-oo?
To determine whether an integral is convergent or divergent without evaluating it, you can use several methods.
For the first integral:
1. Integral -∞ to ∞ [(2x)/(x^2+1)^2] dx:
(a) The integral is improper because it extends to infinity in both directions.
(b) To determine whether it converges or diverges, you need to check the behavior of the integrand as it approaches the limits of integration. In this case, you can analyze the behavior of the function (2x)/(x^2+1)^2 as x approaches ±∞.
As x approaches ±∞, the term x^2 dominates the numerator and denominator. So, the integrand behaves like (2x)/(x^4) = 2/x^3.
Since the exponent of x is greater than 1, the integral converges.
(c) To evaluate the integral, you can use the partial fraction decomposition method or any appropriate method for evaluating integrals of this type.
For the second integral:
2. Integral 0 to π/2 cot(θ) d(θ):
(a) The integral involves an improper integrand because it has an infinity discontinuity at θ = 0.
(b) To determine convergence or divergence, you need to analyze the behavior of the integrand as θ approaches 0.
As θ approaches 0, the cot(θ) term becomes infinite.
Now, you might have made a mistake when taking the limit. If you take the limit of cos(θ)/sin(θ) as θ approaches 0, it becomes undefined. So, the integrand approaches infinity as θ approaches 0.
This indicates that the integral diverges.
Therefore, the integral does not converge, and there is no need to evaluate it further.