What is the integral of \sqrt[3]{x}(sqrt(x)+1) ?

The integral of \sqrt[3]{x}(sqrt(x)+1) is 3/4 x^{4/3}(sqrt(x)+1)-9/44 x^{11/6}+C

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integral of \sqrt[3]{x}(sqrt(x)+1)

\frac{3}{4} x^{\frac{4}{3}} (x + 1) - \frac{9}{44} x^{\frac{11}{6}} + C

Solution steps

Apply Integration By Parts

: \frac{3}{4} x^{\frac{4}{3}} (x + 1) - \int \frac{3}{8} x^{\frac{5}{6}} d x

= \frac{3}{4} x^{\frac{4}{3}} (x + 1) - \int \frac{3}{8} x^{\frac{5}{6}} d x

\int \frac{3}{8} x^{\frac{5}{6}} d x = \frac{9}{44} x^{\frac{11}{6}}

= \frac{3}{4} x^{\frac{4}{3}} (x + 1) - \frac{9}{44} x^{\frac{11}{6}}

Add a constant to the solution

= \frac{3}{4} x^{\frac{4}{3}} (x + 1) - \frac{9}{44} x^{\frac{11}{6}} + C

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What is the integral of \sqrt[3]{x}(sqrt(x)+1) ?
The integral of \sqrt[3]{x}(sqrt(x)+1) is 3/4 x^{4/3}(sqrt(x)+1)-9/44 x^{11/6}+C