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

The integral of (1-x^2)ln(x) is xln(x)-1/3 x^3ln(x)-x+(x^3)/9+C

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integral of (1-x^2)ln(x)

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

x ln (x) - \frac{1}{3} x^{3} ln (x) - x + \frac{x ^{3}}{9} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{3} (3 - x^{2}) d x = x - \frac{x ^{3}}{9}

= ln (x) (x - \frac{x ^{3}}{3}) - (x - \frac{x ^{3}}{9})

Simplify

ln (x) (x - \frac{x ^{3}}{3}) - (x - \frac{x ^{3}}{9}) : x ln (x) - \frac{1}{3} x^{3} ln (x) - x + \frac{x ^{3}}{9}

= x ln (x) - \frac{1}{3} x^{3} ln (x) - x + \frac{x ^{3}}{9}

Add a constant to the solution

= x ln (x) - \frac{1}{3} x^{3} ln (x) - x + \frac{x ^{3}}{9} + C

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What is the integral of (1-x^2)ln(x) ?
The integral of (1-x^2)ln(x) is xln(x)-1/3 x^3ln(x)-x+(x^3)/9+C