What is the integral of (4-x^{2/3})ln(3x) ?

The integral of (4-x^{2/3})ln(3x) is ln(3x)(4x-3/5 x^{5/3})-1/5 (20x-9/5 x^{5/3})+C

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integral of (4-x^{2/3})ln(3x)

\int (4 - x^{\frac{2}{3}}) ln (3 x) d x

ln (3 x) (4 x - \frac{3}{5} x^{\frac{5}{3}}) - \frac{1}{5} (20 x - \frac{9}{5} x^{\frac{5}{3}}) + C

Solution steps

\int (4 - x^{\frac{2}{3}}) ln (3 x) d x

Apply Integration By Parts

: ln (3 x) (4 x - \frac{3}{5} x^{\frac{5}{3}}) - \int \frac{20 x - 3 x ^{\frac{5}{3}}}{5 x} d x

= ln (3 x) (4 x - \frac{3}{5} x^{\frac{5}{3}}) - \int \frac{20 x - 3 x ^{\frac{5}{3}}}{5 x} d x

\int \frac{20 x - 3 x ^{\frac{5}{3}}}{5 x} d x = \frac{1}{5} (20 x - \frac{9}{5} x^{\frac{5}{3}})

= ln (3 x) (4 x - \frac{3}{5} x^{\frac{5}{3}}) - \frac{1}{5} (20 x - \frac{9}{5} x^{\frac{5}{3}})

Add a constant to the solution

= ln (3 x) (4 x - \frac{3}{5} x^{\frac{5}{3}}) - \frac{1}{5} (20 x - \frac{9}{5} x^{\frac{5}{3}}) + C

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What is the integral of (4-x^{2/3})ln(3x) ?
The integral of (4-x^{2/3})ln(3x) is ln(3x)(4x-3/5 x^{5/3})-1/5 (20x-9/5 x^{5/3})+C