What is the integral of xln(2-x) ?

The integral of xln(2-x) is 1/2 x^2ln(2-x)-1/4 x^2-x-2ln|-x+2|+3+C

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integral of xln(2-x)

\int x ln (2 - x) d x

\frac{1}{2} x^{2} ln (2 - x) - \frac{1}{4} x^{2} - x - 2 ln ∣ - x + 2 ∣ + 3 + C

Solution steps

\int x ln (2 - x) d x

Apply Integration By Parts

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

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

\int - \frac{x ^{2}}{2 ( 2 - x )} d x = \frac{1}{4} (x^{2} + 4 x + 8 ln ∣ - x + 2 ∣ - 12)

= \frac{1}{2} x^{2} ln (2 - x) - \frac{1}{4} (x^{2} + 4 x + 8 ln ∣ - x + 2 ∣ - 12)

Expand

- \frac{1}{4} (x^{2} + 4 x + 8 ln ∣ - x + 2 ∣ - 12) : - \frac{1}{4} x^{2} - x - 2 ln ∣ - x + 2 ∣ + 3

= \frac{1}{2} x^{2} ln (2 - x) - \frac{1}{4} x^{2} - x - 2 ln ∣ - x + 2 ∣ + 3

Add a constant to the solution

= \frac{1}{2} x^{2} ln (2 - x) - \frac{1}{4} x^{2} - x - 2 ln ∣ - x + 2 ∣ + 3 + C

\int \frac{1}{3 + sin ^{2} ( x )} d x

f (x) = 7 sin (x) + 3 cos (x)

\frac{1}{2 x}

\frac{d}{d x} (arcsin (1 - x^{2}))

\int \frac{( x ^{3} - 1 )}{x ^{2} ( x - 2 ) ^{3}} d x

What is the integral of xln(2-x) ?
The integral of xln(2-x) is 1/2 x^2ln(2-x)-1/4 x^2-x-2ln|-x+2|+3+C