For this question I am aware I can integrate it normally and get a solution, but is there a specific way required for the reimann improper integral? Do I split it with limits from -1 to 0 and 0 to 2?Also,I think it should be convergent.
2026-05-05 08:55:57.1777971357
Riemann improper integral problem (Introduction to the theory of integration)
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If you uses the widget Definite Integral Calculator from Wolfram Alpha (is free, you can find if type in Google: definite integral calculator, Wolfram alpha), with Integrate: 1/x^2 dx and From: x= -1 to 2, you obtain the message that the integral does not converge, and a visual representation of the integral. I say you this, as a resource that you can freedon of use. Too, you can find many examples of your type of integral, is an integral such that the integrand is discontinuos in an intermediate point $c$. If you type the right words, you will obtain right examples (ask in your mind what words you may type in a search in an internet search engine).
As you said, we can split $\int_{-1}^{2}=\int_{-1}^0+\int_{0}^2$, and use limits to solve the question, but you lost the main tool, the criterion that you uses to solve your problem. Is the criterion in section Discontinuos Integrand Part 3, shown in Example 7 of http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx
Since, for example this one $\int_{0}^2\frac{1}{x^2}dx=\lim_{t\to0^+}\int_{t}^2\frac{1}{x^2}dx=\lim_{t\to 0^+}\left(-\frac{1}{2}+\frac{1}{t}\right)=\infty$, then you can claim that using the cited criterion, your integral $\int_{-1}^2\frac{1}{x^2}dx$ does not converges.
I hope that it is useful for you, too that you write your attempt to solution next time, enco Sorry by my english.
References:
Definite Integral Calculator from Wolfram Alpha, http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605
Notes on Improper integrals: https://cims.nyu.edu/~kiryl/Calculus/Section_6.6--Improper_Integrals/Improper_Integrals.pdf and http://www.math.wisc.edu/~park/Fall2011/integration/Improper%20Integral.pdf