Kerala KEAM 5 June 2024 Solved Paper

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). Quick Tip: For integrals involving powers of $x$ and a sum, use substitution to simplify the expression and integrate each part carefully.