What is the integral of (ln(6x))/(x[ln(3x)]^{-2)} ?

The integral of (ln(6x))/(x[ln(3x)]^{-2)} is 1/3 ln^3(3x)ln(6x)-1/12 ln^4(3x)+C

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integral of (ln(6x))/(x[ln(3x)]^{-2)}

\int \frac{ln ( 6 x )}{x [ ln ( 3 x ) ] ^{- 2}} d x

\frac{1}{3} ln^{3} (3 x) ln (6 x) - \frac{1}{12} ln^{4} (3 x) + C

Solution steps

\int \frac{ln ( 6 x )}{x [ ln ( 3 x ) ] ^{- 2}} d x

Apply Integration By Parts

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

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

\int \frac{ln ^{3} ( 3 x )}{3 x} d x = \frac{1}{12} ln^{4} (3 x)

= \frac{1}{3} ln^{3} (3 x) ln (6 x) - \frac{1}{12} ln^{4} (3 x)

Add a constant to the solution

= \frac{1}{3} ln^{3} (3 x) ln (6 x) - \frac{1}{12} ln^{4} (3 x) + C

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\int \frac{( sin ^{3} ( x ) + 1 )}{sin ^{2} ( x )} d x

What is the integral of (ln(6x))/(x[ln(3x)]^{-2)} ?
The integral of (ln(6x))/(x[ln(3x)]^{-2)} is 1/3 ln^3(3x)ln(6x)-1/12 ln^4(3x)+C