[m] ∫ x^{-\frac{5}{3}}(1+x^{\frac{1}{2}})^{\frac{1}{3}}[/m]
[m]m=-\frac{5}{3}[/m]
[m]n=\frac{1}{2}[/m]
[m]p=\frac{1}{3}[/m]
[m]\frac{m+1}{n}+p=\frac{-\frac{5}{3}+1}{\frac{1}{2}}+\frac{1}{3}=-1[/m]- целое число
Подстановка
[m]x^{-\frac{1}{2}}+1=t^3[/m] ⇒[m] x^{-\frac{1}{2}}=t^3-1[/m] ⇒ [m]x=(t^3-1)^{-2}[/m]
[m]dx=-2\cdot (t^3-1)^{-3}\cdot 3t^2dt[/m]
Тогда
[m] ∫ x^{-\frac{5}{3}}(1+x^{\frac{1}{2}})^{\frac{1}{3}}= ∫(t^3-1)^{\frac{10}{3}}\cdot(\frac{t^3}{t^3-1})^{\frac{1}{3}} \cdot (-2\cdot (t^3-1)^{-3}\cdot 3t^2)dt=-6 ∫t^3dt=-6\cdot \frac{t^4}{4}+C [/m]
[m]t=(x^{-\frac{1}{2}}+1)^{\frac{1}{3}}[/m]
[m]=-\frac{3}{2}(x^{-\frac{1}{2}}+1)^{\frac{4}{3}}+C[/m] - о т в е т