Use the Comparison Theorem to determine whether the following integral is convergent or divergent.
S= integral sign b=infinity and a=1
S(9(cosx^2))/ (1+x^2) dx
well, it is clear that
cosx^2/(1+x^2) <= 1/(1+x^2)
To determine whether the given integral is convergent or divergent, we can use the Comparison Test. Here are the steps to apply the Comparison Test:
1. Identify a known function that is easier to integrate and has properties similar to the given function. In this case, we can compare the given function f(x) = 9(cos(x^2)) / (1 + x^2) with another function g(x) that is easier to integrate.
2. Choose g(x) to be greater than or equal to f(x) for all x in the interval [1, ∞). In this case, we can choose g(x) = 9/(1 + x^2) since it is simpler and clearly greater than or equal to f(x).
3. Integrate g(x) to obtain G(x). In this case, G(x) = 9(arctan(x)).
4. Use the Comparison Test. If G(x) converges, then f(x) converges. If G(x) diverges, then f(x) diverges.
Taking the limit as x approaches infinity of G(x) will determine whether the integral converges or diverges.
lim┬(x→∞)〖G(x)〗 = lim┬(x→∞)9(arctan(x)) = 9(π/2) = ∞
Since the limit is infinity, G(x) diverges. Therefore, the given integral S is also divergent.