Question

$\underset{4}{\overset{9}{\int }}\frac{\sqrt{x}}{{\left(30-x^{3/2}\right)}^2}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_4^9\frac{\sqrt{x}}{{\left(30-x^{{\displaystyle \frac{3}{2}}}\right)}^2}\mathit{d}x\mathit{.}\mathit{}\mathrm{Then}, \\ \mathrm{Let}\left(30-x^{\frac{3}{2}}\right)=t.\mathrm{Then},-\frac{3}{2}\sqrt{x}dx=dt \\ \mathrm{When},x=4,t=22\mathrm{and}x=9,t=3 \\ \therefore I={\int }_{22}^3-\frac{2}{3}\frac{\mathit{1}}{{\mathit{t}}^{\mathit{2}}}\mathit{}dt \\ \Rightarrow I=\frac{2}{3}{\left[\frac{1}{t}\right]}_{22}^3 \\ \Rightarrow I=\frac{2}{3}\left(\frac{1}{3}-\frac{1}{22}\right) \\ \Rightarrow I=\frac{19}{99} \end{gathered}$$