What is the integral of \sqrt[3]{x}ln(x^5) ?

The integral of \sqrt[3]{x}ln(x^5) is 15/4 x^{4/3}ln(x)-45/16 x^{4/3}+C

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integral of \sqrt[3]{x}ln(x^5)

\frac{15}{4} x^{\frac{4}{3}} ln (x) - \frac{45}{16} x^{\frac{4}{3}} + C

Solution steps

Apply Integration By Parts

= \frac{3}{4} x^{\frac{4}{3}} ln (x^{5}) - \int \frac{15}{4} x^{\frac{1}{3}} d x

\int \frac{15}{4} x^{\frac{1}{3}} d x = \frac{45}{16} x^{\frac{4}{3}}

= \frac{3}{4} x^{\frac{4}{3}} ln (x^{5}) - \frac{45}{16} x^{\frac{4}{3}}

\frac{3}{4} x^{\frac{4}{3}} ln (x^{5}) = \frac{15}{4} x^{\frac{4}{3}} ln (x)

= \frac{15}{4} x^{\frac{4}{3}} ln (x) - \frac{45}{16} x^{\frac{4}{3}}

Add a constant to the solution

= \frac{15}{4} x^{\frac{4}{3}} ln (x) - \frac{45}{16} x^{\frac{4}{3}} + C

What is the integral of \sqrt[3]{x}ln(x^5) ?
The integral of \sqrt[3]{x}ln(x^5) is 15/4 x^{4/3}ln(x)-45/16 x^{4/3}+C