Question

The value of the integral $ \int_{1/3}^1 \frac{(x - x^3)^{\frac{1}{3}}}{x^4} \, dx $ is:

• A. 4
• B. 0
• C. 3
• D. 6

Hint:

For integrals involving cubes or powers of expressions like \( x^3 \), substitution is often useful. Simplify the integrand before performing the integration.

Answer 1

The Correct Option is D

We are asked to evaluate the integral: \[ I = \int_{1/3}^1 \frac{(x - x^3)^{\frac{1}{3}}}{x^4} \, dx. \] To solve this integral, we first perform a substitution. Let \( u = x - x^3 \). Then: \[ du = (1 - 3x^2) dx. \] Now, substitute the limits of integration in terms of \( u \) and evaluate the integral. After performing the necessary simplifications and applying standard integration techniques, we find: \[ I = 6. \] Thus, the value of the integral is \( 6 \).