I want to understand intuitively why it seems to be that when evaluating an integral at infinite, it only seems to give a finite value for exponential functions that limit to zero as it approaches infinite. Others such as trigonometric and polynomial functions don't give finite values.(adsbygoogle = window.adsbygoogle || []).push({});

e.g.

[tex]\int_0^{\infty}e^{-x^2}dx=\frac{\sqrt{\pi}}{2}[/tex]

[tex]\int_0^{\infty}\frac{dx}{x}=\lim_{k\to\infty}ln(k)[/tex]

[tex]\int_0^{\pi/2}tan(x)dx=\int_0^{\infty}\left(\frac{\pi}{2}-tan^{-1}x\right)dx=\frac{1}{2}\lim_{k\to\infty}ln(k^2+1)[/tex]

All these functions create the same shape that is, they tend to zero as x tends to infinite, but the difference is that the area under the graphs of the 2nd and 3rd between the x-axis and the function doesn't add to a finite value while it does for the first. How can I understand this intuitively?

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# Integration to infinite

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