What is the integral of x^nln(ax) ?

The integral of x^nln(ax) is 1/(n+1)x^{n+1}ln(ax)-(x^{n+1})/((n+1)^2)+C

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integral of x^nln(ax)

\int x^{n} ln (a x) d x

\frac{1}{n + 1} x^{n + 1} ln (a x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}} + C

Solution steps

\int x^{n} ln (a x) d x

Apply Integration By Parts

: \frac{1}{n + 1} x^{n + 1} ln (a x) - \int \frac{x ^{n}}{n + 1} d x

= \frac{1}{n + 1} x^{n + 1} ln (a x) - \int \frac{x ^{n}}{n + 1} d x

\int \frac{x ^{n}}{n + 1} d x = \frac{x ^{n + 1}}{( n + 1 ) ^{2}}

= \frac{1}{n + 1} x^{n + 1} ln (a x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}}

Add a constant to the solution

= \frac{1}{n + 1} x^{n + 1} ln (a x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}} + C

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What is the integral of x^nln(ax) ?
The integral of x^nln(ax) is 1/(n+1)x^{n+1}ln(ax)-(x^{n+1})/((n+1)^2)+C