I arrived on the below integral as part of solving an ODE, but have no clue how to proceed.
$y = \int\frac{e^x}{x^2}dx$
I tried by parts to reduce it to solving
$y = \int\frac{e^x}{x}dx$
I feel like it is a standard integral, but am unable to proceed.
Would appreciate it if someone could tell me how to search for such things on Google.
On
Two methods to solve integrals with the form $e^x x^{-n} dx$:
Either use the power series of the exponential function, then integrate, then sum back
Do $n$ integration by parts. As you may have noticed, you reduce the order of the exponent of $x$ by one at each integration by parts.
Also please note that $\int e^x x^{-1}dx$ is called the exponential integral and has no standard writing other than $Ei(x)$.
On
The integral $\int \frac{e^x}{x}dx$ cannot be expressed with a finite number of elementary functions.
A closed form requires a special function named Ei$(x)$: http://mathworld.wolfram.com/ExponentialIntegral.html $$\int \frac{e^x}{x}dx=\text{Ei}(x)+\text{constant}.$$
$$\int \frac{e^x}{x^2}dx=\text{Ei}(x)-\frac{e^x}{x}+\text{constant}.$$ If you don't need a closed form one can express it on the form of infinite series (See the above link).
If you are not familiar with the use of special functions see https://en.wikipedia.org/wiki/Special_functions
About special functions, a paper for general public (pp.18-36 in English) : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
The function $\frac{e^x}x$ has no elementary primitive. So, no, it is not a standard integral.