Other Forms for Non-Elementary Integrals

Question

My differential equations textbook has the non-elementary integral...

$$\frac{-1}{4}\int\frac{e^{x}dx}{x}$$

... down as...

$$\frac{-1}{4}\int_{x_0}^x\frac{e^{t}dt}{t}$$ ... in its final solution for a VP problem. All I am wondering is - what is this form of the non-elementary integral called? What is its purpose? Why is it 'more correct' to put it in this form as opposed to the former?

Answer 1

$$\frac{-1}{4}\int_{x_0}^x\frac{e^{t}dt}{t}$$ This defined integral in said "non-elementary" because it cannot be expressed on a closed form with a finite number of so called "elementary functions".

In fact, it is a usual integral for whom familiar with some functions called "special functions", in the present case the function $\text{Ei}(x)$ $$\frac{-1}{4}\int_{x_0}^x\frac{e^{t}dt}{t}=\frac{-1}{4}\left(\text{Ei}(x) -\text{Ei}(x_0)\right)$$

All depends on the background of functions that one knows.

For example, if one doesn't know the function $\text{ln}(x)$, for him the integral $\int \frac{dx}{x}$ is not elementary.

That is the same for one who is confronted with the integral $\int \frac{e^x dx}{x}$ and who doesn't know the function $\text{Ei}(x)$.

An article for the general public about this subject : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales

An alternative way is the use of infinite series. See : http://mathworld.wolfram.com/ExponentialIntegral.html , Eq.(11).

Answer 2

Note that your integral diverges at $0$. Neither form is defined at $x=0$; the most general antiderivative would have two arbitrary constants, one for $x > 0$ and the other for $x < 0$. The second form is only defined for $x$ on the same side of $0$ as $x_0$. It's not "more correct", it's just one particular choice of antiderivative on this side of $0$, namely the one that is $0$ at $x_0$.

Answer 3

You can often see two alternate formulations of the indefinite integral: one is $$ F(x) = \int f(x)\;dx + C $$ where $C$ is a constant;
and the other is $$ F(x) = F(a)+\int_{a}^x f(t)\;dt $$ where $a$ is a point.

For fun, see if you can go from one to the other.