What is the integral of \sqrt[3]{x}ln^2(x) ?

The integral of \sqrt[3]{x}ln^2(x) is 3/4 x^{4/3}ln^2(x)-3/2 (3/4 x^{4/3}ln(x)-9/16 x^{4/3})+C

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integral of \sqrt[3]{x}ln^2(x)

\frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}}) + C

Solution steps

Apply Integration By Parts

: \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x

\int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x = \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}})

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}})

Add a constant to the solution

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}}) + C

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What is the integral of \sqrt[3]{x}ln^2(x) ?
The integral of \sqrt[3]{x}ln^2(x) is 3/4 x^{4/3}ln^2(x)-3/2 (3/4 x^{4/3}ln(x)-9/16 x^{4/3})+C