What is the integral from 2 to 5 of xln(x+2) ?

The integral from 2 to 5 of xln(x+2) is -9/4+21/2 ln(7)

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integral from 2 to 5 of xln(x+2)

\int_{2}^{5} x ln (x + 2) d x

- \frac{9}{4} + \frac{21}{2} ln (7)

Decimal

18.18205 \dots

Solution steps

\int_{2}^{5} x ln (x + 2) d x

Apply u-substitution

: \int_{4}^{7} (u - 2) ln (u) d u

= \int_{4}^{7} (u - 2) ln (u) d u

Apply Integration By Parts

: [ln (u) (\frac{u ^{2}}{2} - 2 u) - \int \frac{1}{2} (u - 4) d u]_{4}^{7}

= [ln (u) (\frac{u ^{2}}{2} - 2 u) - \int \frac{1}{2} (u - 4) d u]_{4}^{7}

\int \frac{1}{2} (u - 4) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 4 u)

= [ln (u) (\frac{u ^{2}}{2} - 2 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 4 u)]_{4}^{7}

Compute the boundaries:

- \frac{9}{4} + \frac{21}{2} ln (7)

= - \frac{9}{4} + \frac{21}{2} ln (7)

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What is the integral from 2 to 5 of xln(x+2) ?
The integral from 2 to 5 of xln(x+2) is -9/4+21/2 ln(7)