integral 1/ (x*e^x) dx

Question

find the integral 1/ (x* e^x ) dx

Answer 1

It is called Ei(x),  exponential integral, and can't be solved by parts or partial fraction analytically.

But it can be solved in power series as shown in the above solution by Chen.

 

For more reference, e^x = 1+ x^1/1! + x^2/2! + x^3/3!+... = Sigma|x=0, infinity| x^n/n!

and e^x/x = 1/x + 1 + x^1/2! + x^2/3! + ...  = Sigma|x=0, infinity|n!/(n+1)!

Therefore,

integral(e^x/x) = ln|x| + x + x^2/(2*2!) + x^3/(3*3!)+ ...  = ln|x| + Sigma|x=1, infinity|x^n/(n*n!) + C

 

 

 

 

 

 

Answer 2

Use McClaurin expansion of e^-x = 1-x+x^2/2-x^3/6+....

to substitute into integral(e^-x/x)dx which can be simplify to integral(1/x - 1 + x/2 - x^2/6 +...)dx which is equal to ln(x) - x + x^2/4 - x^3/18 + ...