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

解答

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

\frac{1}{3} x^{3} ln (x) - x^{2} ln (x) + x ln (x) - \frac{x ^{3}}{9} + \frac{x ^{2}}{2} - x + C

求解步骤

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

使用分布积分法

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

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

= ln (x) (\frac{x ^{3}}{3} - x^{2} + x) - (\frac{x ^{3}}{9} - \frac{x ^{2}}{2} + x)

化简

ln (x) (\frac{x ^{3}}{3} - x^{2} + x) - (\frac{x ^{3}}{9} - \frac{x ^{2}}{2} + x) : \frac{1}{3} x^{3} ln (x) - x^{2} ln (x) + x ln (x) - \frac{x ^{3}}{9} + \frac{x ^{2}}{2} - x

= \frac{1}{3} x^{3} ln (x) - x^{2} ln (x) + x ln (x) - \frac{x ^{3}}{9} + \frac{x ^{2}}{2} - x

解答补常数

= \frac{1}{3} x^{3} ln (x) - x^{2} ln (x) + x ln (x) - \frac{x ^{3}}{9} + \frac{x ^{2}}{2} - x + C

\int x arctan (\frac{2}{x}) d x

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