Mathematics - Indefinite Integration Question with Solution | TestHub

$I=\int \frac{dx}{\sqrt[4]{(x-1{)}^3(x+2{)}^5}}$

$I=\int \frac{dx}{(x-1{)}^{{\displaystyle \frac{3}{4}}}(x+2{)}^{{\displaystyle \frac{5}{4}}}}$

$I=\int \frac{dx}{(x-1{)}^2{\left(\frac{x+2}{x-1}\right)}^{{\displaystyle \frac{5}{4}}}}$

Let $\frac{x+2}{x-1} =t$

Differentiate on both sides

$\frac{\left(x-1\right)-\left(x+2\right)}{{\left(x-1\right)}^2}.dx = dt$

$\Rightarrow \frac{dx}{{\left(x-1\right)}^2} = \frac{-dt}{3}$

$I=\frac{-1}{3}\int \frac{dt}{{\left(t\right)}^{{\displaystyle \frac{5}{4}}}}$

$= \frac{-1}{3}\frac{t^{{\displaystyle \frac{-5}{4}}+1}}{1-{\displaystyle \frac{5}{4}}}+C$

$=\frac{4}{3}{t}^{\frac{-1}{4}}+C$

$=\frac{4}{3}{\left(\frac{1}{t}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+C$

$= \frac{4}{3}{\left(\frac{x-1}{x+2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+C$