What is the integral of x^5ln(2x^3) ?

The integral of x^5ln(2x^3) is 1/6 x^6(ln(2)+3ln(x))-(x^6)/(12)+C

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integral of x^5ln(2x^3)

\int x^{5} ln (2 x^{3}) d x

\frac{1}{6} x^{6} (ln (2) + 3 ln (x)) - \frac{x ^{6}}{12} + C

Solution steps

\int x^{5} ln (2 x^{3}) d x

Apply Integration By Parts

: \frac{1}{6} x^{6} ln (2 x^{3}) - \int \frac{x ^{5}}{2} d x

= \frac{1}{6} x^{6} ln (2 x^{3}) - \int \frac{x ^{5}}{2} d x

\int \frac{x ^{5}}{2} d x = \frac{x ^{6}}{12}

= \frac{1}{6} x^{6} ln (2 x^{3}) - \frac{x ^{6}}{12}

Simplify

ln (2 x^{3}) : 3 ln (x) + ln (2)

= \frac{1}{6} x^{6} (ln (2) + 3 ln (x)) - \frac{x ^{6}}{12}

Add a constant to the solution

= \frac{1}{6} x^{6} (ln (2) + 3 ln (x)) - \frac{x ^{6}}{12} + C

\int \frac{x ^{3} - 2 x ^{2} + x - 1}{x ^{2} - 1} d x

\int cos (m x) cos (n x) d x

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

\int \frac{x}{( 25 - x ^{2} ) ^{\frac{3}{2}}} d x

- \frac{1}{x}

What is the integral of x^5ln(2x^3) ?
The integral of x^5ln(2x^3) is 1/6 x^6(ln(2)+3ln(x))-(x^6)/(12)+C