# A question on the definition of the logarithmic integral

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!

• For instance, li(5) is the integral of 1/ln t with respect to t, from 1 to 5. Sep 6 '14 at 1:56
• @Mehta Thank you. And what is $t$? Sep 6 '14 at 1:57
• I thought the integration range is $1\le t \le x$
– mike
Sep 6 '14 at 1:59
• $t$ is just the variable of integration. It's commonly called a dummy variable, since we could just as well have it be $x',y,Z,\aleph$... Sep 6 '14 at 2:00
• 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 $\displaystyle\int_0^x\frac{dt}{\ln t}$. Sep 6 '14 at 2:29

You can imagine like the definition of natural logarithm $$ln(x)=\int\limits_{1}^{x}\frac{1}{t}dt$$