1.5 Integrals of Mixed Type
Given an integral of mixed type, it can be split into a sum of type one and type two, for example \[\int _0^{\infty }\frac {1}{(1 + x)\, \sqrt {x}\, dx} = \underbrace {\int _0^1\frac {1}{(1 + x)\, \sqrt {x}}\, dx}_{\text {type 2}} + \underbrace {\int _1^{\infty }\frac {1}{(1 + x) \, \sqrt {x}}\, dx}_{\text {type 1}}\]
Exercise 1.5.1. consider the following mixed integrals for convergence or divergence.
- (a).
- \(\displaystyle {\int _0^{\infty } \frac {1}{x\, \sqrt {1 + x^2}}\, dx}\)
- (b).
- \(\displaystyle {\int _0^{\infty } \frac {1}{\sqrt {x}\, \sqrt {1 + x^2}}\, dx}\)
- (c).
- \(\displaystyle {\int _0^{\infty } \frac {t^{-\frac {2}{3}}}{1 + t}\, dt}\)
- (d).
- \(\displaystyle {\int ^{\infty }_0\frac {\log x}{(1 + x^2)^2}}\, dx\)
- (e).
- \(\displaystyle {\int _0^{\pi } \frac {1}{(\sin x)^{\frac {3}{2}}}\, dx}\)
- (f).
- \(\displaystyle {\int _{-\infty }^{\infty } \frac {x}{e^x\, x^4}\, dx}\)
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.