Answer to Question #136202 in Calculus for qwerty

Question

\int \left(a^{\frac{1}{3}}-x^{\frac{1}{3}}\right)^3dx

Answer 1

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

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

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

$=ax-\frac{9}{4}a^{\frac{2}{3}}xx^{\frac{1}{3}}+\frac{9}{5}a^{\frac{1}{3}}xx^{\frac{2}{3}}-\frac{x^4}{4}+C$