the finite case. For instance, numbers 1 and $\G1^2$ give us two different integrals both assuming infinite but different values: $$\int_{0}^{\G1}x^2 dx  = \frac{1}{3}\G1^3,
  \hspace{1cm} \int_{0}^{\G1^2}x^2 dx  = \frac{1}{3}\G1^6.$$ Moreover, it becomes possible to calculate integrals where both endpoints of the interval of integration are infinite as in the following example $$\int_{\G1}^{\G1^2}x^2 dx
  = \frac{1}{3}\G1^6 - \frac{1}{3}\G1^3.$$

We conclude this section by emphasizing that, as it was with sums, it becomes also possible to calculate integrals of functions assuming infinite and infinitesimal values and the obtained results of integration can be finite, infinite, and infinitesimal. For instance, in the integral $$\int_{\G1}^{\G1+\G1^{-2}}x^2 dx
  = \frac{1}{3}(\G1^{1}+\G1^{-2})^3 - \frac{1}{3}\G1^3
  =
  1\G1^{0}+1\G1^{-3}+\frac{1}{3}\G1^{-6}$$ the result has a finite part and two infinitesimal parts and in the integral $$\int_{\G1}^{\G1+\G1^{-2}}{x^2-x\,\,\,} dx
  =
  1\G1^{-3}-\frac{1}{2}\G1^{-4}+\frac{1}{3}\G1^{-6}$$ the result has three infinitesimal parts. The last two examples illustrate situations when the integrand is infinite

$$\int_{\G1}^{\G1^2} \G1 x^2 dx = \G1 \int_{\G1}^{\G1^2} x^2 dx
  = \frac{1}{3}\G1^7 - \frac{1}{3}\G1^4,$$ and infinitesimal $$\int_{\G1}^{\G1^2} \G1^{-4} x^2 dx  = \G1^{-4} \int_{\G1}^{\G1^2}
x^2 dx
  = \frac{1}{3}\G1^{2}  - \frac{1}{3}\G1^{-1}.$$

Methodological consequences and analysis of some\
well known results related to the zeta function {#s4_riem}
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Let us return now to the Riemann zeta function and consider it together 