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The logarithmic integral, which is very important in Number Theory, is defined in the following way:

$$\operatorname{li}(x)=\int_0^x\frac{1}{\ln t}dt\text{.}$$

I don't understand why this equation has two letters or variables ($x$ and $t$). Can someone explain to me what the above equation means? I know what an integral is, but I don't know much calculus.

Thank you!

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  • $\begingroup$ For instance, li(5) is the integral of 1/ln t with respect to t, from 1 to 5. $\endgroup$
    – Mehta
    Sep 6 '14 at 1:56
  • $\begingroup$ @Mehta Thank you. And what is $t$? $\endgroup$
    – User X
    Sep 6 '14 at 1:57
  • $\begingroup$ I thought the integration range is $1\le t \le x$ $\endgroup$
    – mike
    Sep 6 '14 at 1:59
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    $\begingroup$ $t$ is just the variable of integration. It's commonly called a dummy variable, since we could just as well have it be $x',y,Z,\aleph$... $\endgroup$ Sep 6 '14 at 2:00
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    $\begingroup$ All indefinite integrals are definite ones in disguise! Take for instance $\ln x=\displaystyle\int\frac{dx}x$. Yet, open any serious book of mathematics, and you'll find a similar yet slightly different expression: $\displaystyle\int_1^x\frac{dt}t$. Same here: $\text{li}(x)=\displaystyle\int\frac{dx}{\ln x}$ is more correctly written as $\displaystyle\int_0^x\frac{dt}{\ln t}$. $\endgroup$
    – Lucian
    Sep 6 '14 at 2:29
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You can imagine like the definition of natural logarithm $$ln(x)=\int\limits_{1}^{x}\frac{1}{t}dt$$

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