integral of (x^8-x^9)ln(x)

解答

\int (x^{8} - x^{9}) ln (x) d x

\frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100} + C

求解步骤

\int (x^{8} - x^{9}) ln (x) d x

乘开

(x^{8} - x^{9}) ln (x) : x^{8} ln (x) - x^{9} ln (x)

= \int x^{8} ln (x) - x^{9} ln (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{8} ln (x) d x - \int x^{9} ln (x) d x

\int x^{8} ln (x) d x = \frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81}

\int x^{9} ln (x) d x = \frac{1}{10} x^{10} ln (x) - \frac{x ^{10}}{100}

= \frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81} - (\frac{1}{10} x^{10} ln (x) - \frac{x ^{10}}{100})

化简

= \frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100}

解答补常数

= \frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100} + C

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