Establish the facts about the values of the exponent p for which the integral dx

Question

Establish the facts about the values of the exponent p for which the integral dx/x(logx)p is convergent (a>1). Then use either Theorem II or Theorem III to test the following integrals, using the forgoing integral as a standard, with an appropriate value of p in each case. Dx/ x2 + 1[log(1 + x)]2 (x + 1)dx/(x2 - 1)(log x/2 - 1) Let f(x) dx and g(x) dx be two integrals of first kind withy nonnegative integrands, and suppose that f(x) g(x) for all values of x beyond a certain poimt x = c. Then if g(x) dx is convergent, so is f(x) dx, and f(x) dx is divergent, so is g(x) dx. Suppose f(x) dx and g(x) dx are integrals of the first kind with positive integrands, and suppose that the limit f(x)/g(x) = L (22.1 - 3) exists (finite) and is not zero. Then either both integrant, or both are divergent.

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