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We will use the convergence behavior of these improper integrals which have simple integrands to gain understanding of the convergence behavior of improper integrals whose integrands are more complex. For each of these classes, the value of \(p\) or \(a \) will determine if the improper integral converges or diverges. Example 5.103. In this kind of integral one or both of the limits of integration are infinity. In these cases, the interval of integration is said to be over an infinite interval. Let’s take a look at an example that will also show us how we are going to deal with these integrals. Example 1 Evaluate the following integral. ∫ ∞ 1 1 x2 dx ∫ 1 ∞ 1 x 2 d x. test of convergencecomparison tetsimproper integralreal analysis. Here is the idea. Direct Comparison Test for Improper Integrals. Case 1: If on the interval. and converges. and if we have a second function which is also on. and we know that for all in the interval , then we can conclude that also converges. Case 2: If on the interval. and diverges. An improper integral of type 1 is an integral whose interval of integration is infinite . This means the limits of integration include ∞ or − ∞ or both . Remember that ∞ is a process (keep going and never stop), not a number. Therefore, we cannot use ∞ as an actual limit of integration as in the FTC II . We make the limit of. Theorem 2 (Absolute convergence implies convergence.). If the improper integral (1) con-verges absolutely then it converges. Proof. We make use of the Cauchy criterion. Let >0. Since the improper integral of jf(x)j converges we can nd an M aso that for all A;B Mwe have Z B A jf(x)jdx < : But the integral of jf(x)jis nonnegative, so we have Z B.