Steps in Resolving the Integral: $\int\frac{1}{x^3}e^\frac{1}{x}dx$

Question

Could anyone assist me in a step by step solution to solving the following integral?

$$ \int\frac{1}{x^3}e^\frac{1}{x}\,dx $$

I have tried using both sides in integration by parts, but can't seem to come to a solution.

Answer 1

Take $u = \frac{1}{x}, du = -\frac{1}{x^{2}}$. Therefore: $ \large\int\frac{1}{x^3}e^\frac{1}{x}dx = \int -ue^{u}du$. Now use parts.

Answer 2

$$\int \frac { 1 }{ x^{ 3 } } e^{ \frac { 1 }{ x } }dx=-\int \frac { 1 }{ x } e^{ \frac { 1 }{ x } }d\left( \frac { 1 }{ x } \right) \\$$

then set $$ \frac { 1 }{ x } =t$$ so that to solve $$-\int { t{ e }^{ t }dt } $$ by parts