Question

\[ \int_{0}^{8} x^{\frac{5}{3}} \left(4 - x^{\frac{2}{3}}\right)^{\frac{3}{2}} \, dx = \ ? \] 

• A. \(\frac{2^{16}}{385}\)
• B. \(\frac{2^{13}}{385}\)
• C. \(\frac{2^{12}}{385}\)
• D. \(\frac{2^{15}}{385}\) \bigskip

Hint:

When powers are fractional, try \(x=t^n\) substitution first.

Answer 1

The Correct Option is A

Concept: Substitute \(x=t^3\).

Step 1: Substitution.
\[ x=t^3 \Rightarrow dx=3t^2 dt \] \[ x^{5/3}=t^5,\quad x^{2/3}=t^2 \]

Step 2: Transform integral.
\[ I=\int_0^2 3t^7(4-t^2)^{3/2} dt \]

Step 3: Beta function form.
Let \(t=2\sin\theta\) \[ I=\frac{2^{16}}{385} \] \[ \boxed{(A)} \]