Question

Under the substitution \[30-x^{\frac32}=t,\] what is the correct expression for \(\sqrt{x}\,dx\)?

Options

• \[\sqrt{x}\,dx=-\frac23\,dt\]

• \[\sqrt{x}\,dx=\frac23\,dt\]

• \[\sqrt{x}\,dx=2\,dt\]

• \[\sqrt{x}\,dx=-\frac32\,dt\]

Answer 1

Differentiating \(30-x^{\frac32}=t\) gives \(-\frac32\sqrt{x}\,dx=dt\).
Solving for \(\sqrt{x}\,dx\) gives \(-\frac23\,dt\).