What is the integral of log_{3}(6x^2)x ?

The integral of log_{3}(6x^2)x is 1/2 log_{3}(6)x^2+x^2log_{3}(x)-(x^2)/(2ln(3))+C

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integral of log_{3}(6x^2)x

\int lo g_{3} (6 x^{2}) x d x

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

Solution steps

\int lo g_{3} (6 x^{2}) x d x

Apply Integration By Parts

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

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

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

Simplify

\frac{1}{2} x^{2} lo g_{3} (6 x^{2}) - \frac{x ^{2}}{2 ln ( 3 )} : \frac{1}{2} lo g_{3} (6) x^{2} + x^{2} lo g_{3} (x) - \frac{x ^{2}}{2 ln ( 3 )}

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

Add a constant to the solution

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

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