What is the integral of x^9ln(3x^5) ?

The integral of x^9ln(3x^5) is 1/10 x^{10}(ln(3)+5ln(x))-(x^{10})/(20)+C

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integral of x^9ln(3x^5)

\int x^{9} ln (3 x^{5}) d x

\frac{1}{10} x^{10} (ln (3) + 5 ln (x)) - \frac{x ^{10}}{20} + C

Solution steps

\int x^{9} ln (3 x^{5}) d x

Apply Integration By Parts

: \frac{1}{10} x^{10} ln (3 x^{5}) - \int \frac{x ^{9}}{2} d x

= \frac{1}{10} x^{10} ln (3 x^{5}) - \int \frac{x ^{9}}{2} d x

\int \frac{x ^{9}}{2} d x = \frac{x ^{10}}{20}

= \frac{1}{10} x^{10} ln (3 x^{5}) - \frac{x ^{10}}{20}

Simplify

ln (3 x^{5}) : 5 ln (x) + ln (3)

= \frac{1}{10} x^{10} (ln (3) + 5 ln (x)) - \frac{x ^{10}}{20}

Add a constant to the solution

= \frac{1}{10} x^{10} (ln (3) + 5 ln (x)) - \frac{x ^{10}}{20} + C

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What is the integral of x^9ln(3x^5) ?
The integral of x^9ln(3x^5) is 1/10 x^{10}(ln(3)+5ln(x))-(x^{10})/(20)+C