What is the integral of xln(5-x) ?

The integral of xln(5-x) is 1/2 x^2ln(5-x)+1/2 (-(x^2)/2-5x-25ln|-x+5|)+C

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integral of xln(5-x)

\int x ln (5 - x) d x

\frac{1}{2} x^{2} ln (5 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

Solution steps

\int x ln (5 - x) d x

Apply Integration By Parts

= \frac{1}{2} x^{2} ln (5 - x) - \int - \frac{x ^{2}}{2 ( 5 - x )} d x

\int - \frac{x ^{2}}{2 ( 5 - x )} d x = - \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣)

= \frac{1}{2} x^{2} ln (5 - x) - (- \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣))

Simplify

= \frac{1}{2} x^{2} ln (5 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣)

Add a constant to the solution

= \frac{1}{2} x^{2} ln (5 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

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What is the integral of xln(5-x) ?
The integral of xln(5-x) is 1/2 x^2ln(5-x)+1/2 (-(x^2)/2-5x-25ln|-x+5|)+C