What is the integral of ((e^{1/x}+2))/(x^2) ?

The integral of ((e^{1/x}+2))/(x^2) is -e^{1/x}-2/x+C

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integral of ((e^{1/x}+2))/(x^2)

\int \frac{( e ^{\frac{1}{x}} + 2 )}{x ^{2}} d x

- e^{\frac{1}{x}} - \frac{2}{x} + C

Solution steps

\int \frac{e ^{\frac{1}{x}} + 2}{x ^{2}} d x

Apply u-substitution

: \int - e^{u} - 2 d u

= \int - e^{u} - 2 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int e^{u} d u - \int 2 d u

\int e^{u} d u = e^{u}

\int 2 d u = 2 u

= - e^{u} - 2 u

Substitute back

u = \frac{1}{x}

= - e^{\frac{1}{x}} - 2 \cdot \frac{1}{x}

2 \cdot \frac{1}{x} = \frac{2}{x}

= - e^{\frac{1}{x}} - \frac{2}{x}

Add a constant to the solution

= - e^{\frac{1}{x}} - \frac{2}{x} + C

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What is the integral of ((e^{1/x}+2))/(x^2) ?
The integral of ((e^{1/x}+2))/(x^2) is -e^{1/x}-2/x+C