What is the integral of (3x^8-x^9)ln(x) ?

The integral of (3x^8-x^9)ln(x) is 1/3 x^9ln(x)-(x^9)/(27)-1/10 x^{10}ln(x)+(x^{10})/(100)+C

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integral of (3x^8-x^9)ln(x)

\int (3 x^{8} - x^{9}) ln (x) d x

\frac{1}{3} x^{9} ln (x) - \frac{x ^{9}}{27} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100} + C

Solution steps

\int (3 x^{8} - x^{9}) ln (x) d x

Expand

(3 x^{8} - x^{9}) ln (x) : 3 x^{8} ln (x) - x^{9} ln (x)

= \int 3 x^{8} ln (x) - x^{9} ln (x) d x

Apply the Sum Rule:

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

= \int 3 x^{8} ln (x) d x - \int x^{9} ln (x) d x

\int 3 x^{8} ln (x) d x = 3 (\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}

= 3 (\frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81}) - (\frac{1}{10} x^{10} ln (x) - \frac{x ^{10}}{100})

Simplify

3 (\frac{1}{9} x^{9} ln (x) - \frac{x ^{9}}{81}) - (\frac{1}{10} x^{10} ln (x) - \frac{x ^{10}}{100}) : \frac{1}{3} x^{9} ln (x) - \frac{x ^{9}}{27} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100}

= \frac{1}{3} x^{9} ln (x) - \frac{x ^{9}}{27} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100}

Add a constant to the solution

= \frac{1}{3} x^{9} ln (x) - \frac{x ^{9}}{27} - \frac{1}{10} x^{10} ln (x) + \frac{x ^{10}}{100} + C

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What is the integral of (3x^8-x^9)ln(x) ?
The integral of (3x^8-x^9)ln(x) is 1/3 x^9ln(x)-(x^9)/(27)-1/10 x^{10}ln(x)+(x^{10})/(100)+C