Question

\[ \int \left(27x^3(1-x^3)\right)^{\frac{2}{3}}\,dx = \]

• A. \(-\frac{3}{4}(1 - x^3)^{4/3} + C\)
• B. \(-\frac{3}{5}(1 - x^3)^{5/3} + C\)
• C. \(-\frac{9}{3}(1 - x^3)^{4/3} + C\)
• D. \(-\frac{9}{4}(1 - x^3)^{4/3} + C\)
• E. \(-\frac{9}{5}(1 - x^3)^{5/3} + C\)

Hint:

Always simplify constant coefficients like \(27^{2/3}\) first. It makes the substitution steps much cleaner.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We first distribute the exponent and simplify the expression, then look for a substitution.

Step 2: Key Formula or Approach:
Distribute power: \((ab)^n = a^n b^n\).
\(27^{2/3} = (3^3)^{2/3} = 3^2 = 9\).

Step 3: Detailed Explanation:
The integrand is:
\[ 27^{2/3} \cdot (x^3)^{2/3} \cdot (1 - x^3)^{2/3} = 9 x^2 (1 - x^3)^{2/3} \]
Now use substitution:
Let \(u = 1 - x^3\). Then \(du = -3x^2 dx \implies x^2 dx = -\frac{du}{3}\).
The integral becomes:
\[ \int 9 \cdot u^{2/3} \cdot \left( -\frac{du}{3} \right) = -3 \int u^{2/3} du \]
Integrate:
\[ = -3 \cdot \frac{u^{5/3}}{5/3} + C = -3 \cdot \frac{3}{5} u^{5/3} + C = -\frac{9}{5}(1 - x^3)^{5/3} + C \]

Step 4: Final Answer:
The value is \(-\frac{9}{5}(1 - x^3)^{5/3} + C\).