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

The integral of x^2ln(sqrt(1-x)) is 1/2 (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(sqrt(1-x))

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

\frac{1}{2} (\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

Simplify

ln (1 - x) : \frac{1}{2} ln (- x + 1)

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

Take the constant out:

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

= \frac{1}{2} \cdot \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

= \frac{1}{2} (\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}{2} (\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}{2} (\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}{2} (\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(sqrt(1-x)) ?
The integral of x^2ln(sqrt(1-x)) is 1/2 (1/3 x^3ln(1-x)+1/3 (-(x^3)/3-(x^2)/2-x-ln|-x+1|))+C