What is the integral from 3 to 4 of-xln(x-3) ?

The integral from 3 to 4 of-xln(x-3) is 13/4

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integral from 3 to 4 of-xln(x-3)

\int_{3}^{4} - x ln (x - 3) d x

\frac{13}{4}

Decimal Notation

3.25

Solution steps

\int_{3}^{4} - x ln (x - 3) d x

Take the constant out:

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

= - \int_{3}^{4} x ln (x - 3) d x

Apply u-substitution

: \int_{0}^{1} (u + 3) ln (u) d u

= - \int_{0}^{1} (u + 3) ln (u) d u

Apply Integration By Parts

: [ln (u) (\frac{u ^{2}}{2} + 3 u) - \int \frac{1}{2} (u + 6) d u]_{0}^{1}

= - [ln (u) (\frac{u ^{2}}{2} + 3 u) - \int \frac{1}{2} (u + 6) d u]_{0}^{1}

\int \frac{1}{2} (u + 6) d u = \frac{1}{2} (\frac{u ^{2}}{2} + 6 u)

= - [ln (u) (\frac{u ^{2}}{2} + 3 u) - \frac{1}{2} (\frac{u ^{2}}{2} + 6 u)]_{0}^{1}

Compute the boundaries:

- \frac{13}{4}

= - (- \frac{13}{4})

Simplify

= \frac{13}{4}

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What is the integral from 3 to 4 of-xln(x-3) ?
The integral from 3 to 4 of-xln(x-3) is 13/4