What is the integral of x^2ln(1-x) ?

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

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integral of x^2ln(1-x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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