I cannot solve the integration

Question

Please help me to solve the integration below:
$$\int (e^{x - 1/x})(1+1/x^2) \mathrm dx$$

There must be some typo as $$\dfrac{d(x\pm 1/x)}{dx}=1\mp1/x^2$$ — lab bhattacharjee, Jan 04 '16 at 14:24
Unless there is a typo as lab bhattacharjee says, the integral can't be found in terms of elementary functions, according to WA. — zz20s, Jan 04 '16 at 14:47
This integral cannot be computed. Are you sure it's not supposed to be $$\int \left(e^{x+\frac{1}{x}}\right)\left(1-\frac{1}{x^2}\right)\text{d}x\quad\text{ or };;;\int\left(e^{x-\frac{1}{x}}\right)\left(1+\frac{1}{x^2}\right)\text{d}x;\text{?}$$ — Corellian, Jan 04 '16 at 14:55

score 1 · Accepted Answer · answered Jan 04 '16 at 15:12

1

Let $u=x-\frac{1}{x}$. Thus, $\mathrm du=1+\frac{1}{x^2} \mathrm dx$. Now the integral is $$\int e^u \mathrm du=e^u+C=e^{x-\frac{1}{x}}+C$$

This is easily checked by differentiation of $e^{x-1/x}.$

answered Jan 04 '16 at 15:12

zz20s

1 Answers1