Question

An antiderivative of \[\frac{\sqrt{x}}{(30-x^{\frac32})^2}\] is:

Options

• \[\frac{2}{3}\left[\frac{1}{30-x^{\frac32}}\right]\]

• \[\frac{1}{(30-x^{\frac32})^2}\]

• \[\frac{2}{3}(30-x^{\frac32})\]

• \[-\frac{2}{3}\left[\frac{1}{30-x^{\frac32}}\right]\]

Answer 1

Using \(t=30-x^{\frac32}\), the integral becomes \(-\frac23\int t^{-2}\,dt\).
This evaluates to \(\frac23\left[\frac{1}{30-x^{\frac32}}\right]\).