Question

integrate[ e raised to power x] ÷ x wrt dx

Answer 1

This integral is called Exponential integral function. We can not find a simple answer in terms of known functions. One main problem is that this function has a singularity at x = 0, where it is not defined.

So we expand  e^x as a Taylor series expansion at x = 0 and then integrate.

e^x = 1 + x + x² /2! + x³ /3! + ......
e^x / x = 1/x + 1 + x/2! + x² /3!  + x³ / 4! + .....       FOR  x ≠ 0

Now we perform integration:

$Ei(x) = \gamma+ \int {\frac{dx}{x}}+\int {1} dx+\int {\frac{x}{2!}} dx+\int {\frac{x^2}{3!}} dx....+\int {\frac{x^{n-1}}{n!}} dx+...\\\\Ei(x)=\gamma+Ln|x| +x+\frac{x^2}{2*2!}+\frac{x^3}{3*3!}+...+\frac{x^n}{n*n!}+...\\\\\ \ where\ \gamma=a\ mathematical\ constant$