integral If f(x) dr exists for every number t > a, we can define an improper f(x) delim [f(x) dr. If f0, then this integral corresponds to the area of the unbounded region enclosed by f and the x-axis from a to infinity, shown below. S y= f(x) We say that the improper integral f(x) dx is convergent if this limit exists, otherwise we call it divergent. Similarly, if f(x) is continuous on (a, b] and discontinuous or undefined at a (in par- ticular, if x = a is a vertical asymptote), as in the graph below, 0 at bx we define an improper integral 1 f(x) dx = lim f(x) dr. tat If 0, then this integral corresponds to the area of the unbounded region enclosed by f and the x-axis from a to b. Again, we say f(x) dr is convergent if this limit exists, otherwise we call it divergent. Note that there is more information on this topic, including fully-solved examples in Section 11.7 Improper Integrals. In this question, we are going to focus on the integral dx This integral is potentially improper in two distinct ways: the interval of integration is unbounded and x = 0 is (depending on the value of p) a vertical asymptote. To investigate this integral, we will rewrite it as the sum of two integrals and evaluate each independently. dx = dx + -~-~ dx (a) For what values of p is the following integral convergent? 6 dx (b) For what values of p is the following integral convergent? dx (c) If each of these integrals described in parts (a) and (b) converge with values L and L2, then the original integral converges and has value L+ L2. Otherwise, the original integral is divergent. For what values of p is the original integral fodx convergent?