∫(x23−1x23+2x5−2x)dx=∫x23dx−∫x−23dx+2∫x−5dx−2∫x1dx=\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=∫(x32−x321+x52−2x)dx=∫x32dx−∫x−32dx+2∫x−5dx−2∫x1dx=
=x23+123+1−x−23+1−23+1+2x−5+1−5+1−2x1+11+1+C==\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==32+1x32+1−−32+1x−32+1+2−5+1x−5+1−21+1x1+1+C=
=3x535−3x13+2x−4−4−2x22+C==\frac{3x^{\frac{5}{3}}}{5}-3x^{\frac{1}{3}}+2\frac{x^{-4}}{-4}-2\frac{x^2}{2}+C==53x35−3x31+2−4x−4−22x2+C=
=35x53−3x3−12x4−x2+C=\frac{3}{5}{x^{\frac{5}{3}}}-3\sqrt[3]{x}-\frac{1}{2x^4}-x^2+C=53x35−33x−2x41−x2+C.
Answer: 35x53−3x3−12x4−x2+C\frac{3}{5}{x^{\frac{5}{3}}}-3\sqrt[3]{x}-\frac{1}{2x^4}-x^2+C53x35−33x−2x41−x2+C.
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