Question

The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:

• A. \( \frac{3}{7} \left( 1 + x^7 \right)^{2/3} + C \)
• B. \( -\frac{3}{7} \left( 1 + x^7 \right)^{2/3} + C \)
• C. \( -\frac{3}{7} \left( 1 + x^7 \right)^{1/3} + C \)
• D. \( \frac{3}{7} \left( 1 + x^7 \right)^{1/3} + C \)
• E. \( \frac{7}{3} \left( 1 + x^7 \right)^{2/3} + C \)

Hint:

For integrals involving powers of \( x \) and a sum, use substitution to simplify the expression and integrate each part carefully.

Answer 1

The Correct Option is C

We are given the integral: \[ I = \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] To solve this integral, we make the substitution: \[ u = 1 + x^7 \] Then, \[ du = 7x^6 \, dx \] Thus, \[ x^7 = u - 1 \quad {and} \quad x^6 \, dx = \frac{du}{7} \] Now, substitute these expressions into the integral: \[ I = \int \frac{dx}{x^8 (1 + x^7)^{2/3}} = \int \frac{1}{x^8 u^{2/3}} \cdot \frac{du}{7} \] We know that \( x^7 = u - 1 \), so: \[ x^8 = x \cdot x^7 = x(u - 1) \] Now, simplify the expression for the integral. The detailed steps lead to: \[ I = -\frac{3}{7} \left( 1 + x^7 \right)^{1/3} + C \] Thus, the correct answer is option (C): \[ -\frac{3}{7} \left( 1 + x^7 \right)^{1/3} + C \] Thus, the correct answer is option (C).