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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{e ^{u}}{1 + e ^{u}} d u

= \int - \frac{e ^{u}}{1 + e ^{u}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{e ^{u}}{1 + e ^{u}} d u

Apply u-substitution

: \int \frac{1}{v} d v

= - \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln 1 + e^{\frac{1}{x}}

Add a constant to the solution

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

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