Question

\[\int\limits_4^9 \frac{\sqrt{x}}{\left( 30 - x^{3/2} \right)^2} dx\]

Answer 1

\[Let\ I = \int_4^9 \frac{\sqrt{x}}{\left( 30 - x^\frac{3}{2} \right)^2} d x . Then, \]
\[Let \left( 30 - x^\frac{3}{2} \right) = t . Then, - \frac{3}{2}\sqrt{x} dx = dt\]
\[When\, x = 4, t = 22\ and\ x\ = 9, t = 3\]
\[ \therefore I = \int_{22}^3 - \frac{2}{3}\frac{1}{t^2} dt\]
\[ \Rightarrow I = \frac{2}{3} \left[ \frac{1}{t} \right]_{22}^3 \]
\[ \Rightarrow I = \frac{2}{3}\left( \frac{1}{3} - \frac{1}{22} \right)\]
\[ \Rightarrow I = \frac{19}{99}\]