Question

\int \:\left(x^{\frac{2}{3}}-\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^5}-2x\right)dx

Answer 1

$\int \left(x^{\frac{2}{3}}-\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^5}-2x\right)dx=\int x^{\frac{2}{3}}dx-\int x^{-\frac{2}{3}}dx+2\int x^{-5}dx-2\int x^1dx=$

$=\frac{x^{{\frac{2}{3}}+1}}{\frac{2}{3}+1} -\frac{x^{-{\frac{2}{3}}+1}}{-\frac{2}{3}+1} +2\frac{x^{-5+1}}{-5+1} -2\frac{x^{1+1}}{1+1}+C=$

$=\frac{3x^{\frac{5}{3}}}{5}-3x^{\frac{1}{3}}+2\frac{x^{-4}}{-4}-2\frac{x^2}{2}+C=$

$=\frac{3}{5}{x^{\frac{5}{3}}}-3\sqrt[3]{x}-\frac{1}{2x^4}-x^2+C$.

Answer: $\frac{3}{5}{x^{\frac{5}{3}}}-3\sqrt[3]{x}-\frac{1}{2x^4}-x^2+C$.