\(2)\). The integral \(\displaystyle \int_{a}^{\infty} \frac{dx}{(x)^k}\) where a>0, is convergent iff \(k>1\).
\(3)\). The integral \(\displaystyle \int_{1}^{\infty}\frac{x^{-k}}{1+x}dx\), is convergrnt provided \(k>0\).
\(4)\). The integral \(\displaystyle \int_{a}^{b} \frac{dx}{(b-x)^k}\) is convergent iff \(k<1 br="">
\(5)\). Cauchy's integral test: If for \(x \geq 1\), \(f(x)\) is a non-negative monotonicallyu decreasing integrable function of \(x\) such that \(f(n) = u_n\) for all positive integral value of \(n,\) then the series \(\displaystyle \sum_{n=1}^{\infty} u_n\) and the improper integral \(\displaystyle \int_{1}^{\infty} f(x)dx\) converges or diverges togather.
\(6)\). Comparison test : Let \(\displaystyle \int_{a}^{b} f(x) dx\) and \(\displaystyle \int_{a}^{b} g(x) dx\) be two improper integrals, where both are integrable in \([a+\epsilon,b]\) s.t \(0< \epsilon < b-a\)
(i) \('a'\) is the only point of infinite discontinuity of \(f\) and \(g\).
(ii) \(f\) and \(g\) be two non-negative functions
(iii) \(f \leq kg \;\; \forall x \in (a,b]\;\;\) where \(\;k >0 \;\)then
\(\displaystyle \int_{a}^{b} f(x) dx\;\;\) converges if \(\displaystyle \int_{a}^{b} g(x)dx\;\;\) converges
\(\displaystyle \int_{a}^{b} g(x) dx\;\;\) diverges if \(\displaystyle \int_{a}^{b}f(x)dx\;\;\) diverges
\(7)\). Limit Comparison test : Let \(\displaystyle \int_{a}^{b} f(x) dx\) and \(\displaystyle \int_{a}^{b} g(x) dx\) be two improper integrals, where
(i) \('a'\) is the only point of infinite discontinuity of \(f\) and \(g\).
(ii) \(f\) and \(g\) be two positive functions
if \(\lim_{x \to a+} \frac{f(x)}{g(x)} = l\;\;\) where \(l\) is non zero finite number, both integral then converges or diverges together.
\(8)\). \(\displaystyle \int_{0}^{\frac{\pi}{2}} \log \sin x \;dx\;\) is convergent.
\(9)\). \(\displaystyle \int_{0}^{1} \frac{\sin(1/x)}{\sqrt{x}}\; dx\;\) is convergent.
\(10)\). \(\displaystyle \int_{0}^{\infty} \frac{\sin x}{x}\; dx\;\) is convergent.
\(11).\) \(\displaystyle \int_{0}^{\infty} \left|\frac{\sin x}{x}\right| dx\;\) is divergent.
\(12)\). \(\displaystyle \int_{0}^{\infty} \frac{\sin nx}{x}\; dx\;= \frac{\pi}{2}\;\;\) or \(\displaystyle \frac{-\pi}{2}\) according to n > 0 or n < 0 repectively is convergent.
\(13)\). \(\displaystyle \int_{0}^{\infty} x^k(\log (x))^n\; dx\;\) is convergent if k < -1
\(14).\) \(\displaystyle \int_{0}^{\infty} \frac{x^k(1+x)^n}{1+x^m} dx\;\) (where k and n are positive) is convergent if \(\;m > 1+k+n\).
Examples of some absolutily convergent improper integrals
\(1)\). \(\displaystyle \int_{0}^{\infty} \frac{\sin x}{1+x^2}\; dx\;\) is convergent.
\(2).\) \(\displaystyle \int_{0}^{\infty} \frac{\cos x}{1+x^2} dx\;\) is convergent.
\(3)\). \(\displaystyle \int_{0}^{\infty} \frac{\cos ax}{e^{mx}}\; dx\;\) is convergent where \(m>0\).
\(4)\). \(\displaystyle \int_{0}^{\infty} \frac{\sin x}{\sqrt{x+x^3}}\; dx\;\) is convergent.
\(5).\) \(\displaystyle \int_{0}^{\infty} \frac{x\sin x}{1+x^3} dx\;\) is convergent.
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