Question

Evaluate the integral: \[ \int \left( 27x^3 (1 - x^3)^{\frac{1}{3}} \right) dx \]

• A. \( \frac{27}{4} (1 - x^3)^{\frac{4}{3}} + C \)
• B. \( \frac{27}{5} (1 - x^3)^{\frac{5}{3}} + C \)
• C. \( -\frac{27}{4} (1 - x^3)^{\frac{4}{3}} + C \)
• D. \( -\frac{27}{5} (1 - x^3)^{\frac{5}{3}} + C \)

Hint:

When performing substitution, always remember to change both the differential and the integrand in terms of the new variable. This simplifies the integral.

Answer 1

The Correct Option is C

We are tasked with evaluating the integral: \[ \int 27x^3 (1 - x^3)^{\frac{1}{3}} \, dx \] Step 1: Use substitution.
Let’s perform substitution to simplify the expression. Let: \[ u = 1 - x^3 \] Differentiating both sides with respect to \( x \), we get: \[ du = -3x^2 \, dx \] Now, express \( x^3 \, dx \) in terms of \( du \): \[ x^3 \, dx = -\frac{1}{3} \, du \]
Step 2: Substitute in the integral.
Substitute \( x^3 \, dx \) and \( (1 - x^3)^{\frac{1}{3}} \) in terms of \( u \): \[ \int 27x^3 (1 - x^3)^{\frac{1}{3}} \, dx = \int 27 \left( -\frac{1}{3} \, du \right) u^{\frac{1}{3}} \] Simplifying the constants: \[ = -9 \int u^{\frac{1}{3}} \, du \]
Step 3: Integrate.
Now, integrate \( u^{\frac{1}{3}} \): \[ \int u^{\frac{1}{3}} \, du = \frac{3}{4} u^{\frac{4}{3}} \] So, the integral becomes: \[ -9 \times \frac{3}{4} u^{\frac{4}{3}} = -\frac{27}{4} u^{\frac{4}{3}} \]
Step 4: Substitute \( u \) back in terms of \( x \).
Substitute \( u = 1 - x^3 \) back into the expression: \[ -\frac{27}{4} (1 - x^3)^{\frac{4}{3}} + C \] Thus, the solution to the integral is: \[ \boxed{-\frac{27}{4} (1 - x^3)^{\frac{4}{3}} + C} \]