What is the integral from 2 to 3 of (x-1)/(x^2)e^x ?

The integral from 2 to 3 of (x-1)/(x^2)e^x is (e^3)/3-(e^2)/2

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integral from 2 to 3 of (x-1)/(x^2)e^x

\int_{2}^{3} \frac{x - 1}{x ^{2}} e^{x} d x

\frac{e ^{3}}{3} - \frac{e ^{2}}{2}

Decimal

3.00065 \dots

Solution steps

\int_{2}^{3} \frac{x - 1}{x ^{2}} e^{x} d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int_{2}^{3} \frac{e ^{x} ( x - 1 )}{x ^{2}} d x

Apply Integration By Parts

: [- \frac{e ^{x} ( x - 1 )}{x} - \int - e^{x} d x]_{2}^{3}

= [- \frac{e ^{x} ( x - 1 )}{x} - \int - e^{x} d x]_{2}^{3}

\int - e^{x} d x = - e^{x}

= [- \frac{e ^{x} ( x - 1 )}{x} - (- e^{x})]_{2}^{3}

Simplify

[- \frac{e ^{x} ( x - 1 )}{x} - (- e^{x})]_{2}^{3} : [\frac{e ^{x}}{x}]_{2}^{3}

= [\frac{e ^{x}}{x}]_{2}^{3}

Compute the boundaries:

\frac{e ^{3}}{3} - \frac{e ^{2}}{2}

= \frac{e ^{3}}{3} - \frac{e ^{2}}{2}

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What is the integral from 2 to 3 of (x-1)/(x^2)e^x ?
The integral from 2 to 3 of (x-1)/(x^2)e^x is (e^3)/3-(e^2)/2