Which of the following best describes the function $E i (x)$ ?

It is the exponential integral, defined as

E i (x) = \int_{- \infty}^{x} \frac{e^{t}}{t} d t

It is the error function, defined as

e r f (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t

It is the natural exponential function, defined as

e^{x}

It is the inverse of the exponential function, defined as

l n (x)

Verified step by step guidance

Step 1: Understand the problem by identifying the function in question, which is $\operatorname{Ei}$(x). This is referred to as the exponential integral.

Step 2: Recall the definition of the exponential integral $\operatorname{Ei}$(x). It is defined mathematically as $\operatorname{Ei}$(x) = $\int$_{-$\infty$}^x $\frac{e^t}{t}$ dt.

Step 3: Compare the given options to the definition of $\operatorname{Ei}$(x). The first option matches the definition of the exponential integral, $\operatorname{Ei}$(x).

Step 4: Review the other options to ensure they do not describe $\operatorname{Ei}$(x). For example, $\operatorname{erf}$(x) is the error function, e^x is the natural exponential function, and $\ln$(x) is the natural logarithm, which is the inverse of the exponential function.

Step 5: Conclude that the correct description of $\operatorname{Ei}$(x) is the first option: 'It is the exponential integral, defined as $\operatorname{Ei}$(x) = $\int$_{-$\infty$}^x $\frac{e^t}{t}$ dt.'

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