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

The integral from 1 to 2 of 1/(x^2)e^{1/x} is e-sqrt(e)

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

\int_{1}^{2} \frac{1}{x ^{2}} e^{\frac{1}{x}} d x

e - e

Decimal

1.06956 \dots

Solution steps

\int_{1}^{2} \frac{1}{x ^{2}} e^{\frac{1}{x}} d x

Apply u-substitution

: \int_{1}^{\frac{1}{2}} - e^{u} d u

= \int_{1}^{\frac{1}{2}} - e^{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{2}}^{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{1}{2}}^{1} e^{u} d u)

Use the common integral:

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

= - (- [e^{u}]_{\frac{1}{2}}^{1})

Simplify

= [e^{u}]_{\frac{1}{2}}^{1}

Compute the boundaries:

e - e

= e - e

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What is the integral from 1 to 2 of 1/(x^2)e^{1/x} ?
The integral from 1 to 2 of 1/(x^2)e^{1/x} is e-sqrt(e)