Question

If $ f(x) = \frac{x}{(1 + nx^n)^{1/n}} $ for $ n \geq 2 $, then \[ \int x^{n-2} f(x) \, dx = \]

• A. \( \frac{1}{n(n-1)}(1 + nx^n)^{1 - \frac{1}{n}} + C \)
• B. \( \frac{1}{n-1}(1 + nx^n)^{1 - \frac{1}{n}} + C \)
• C. \( \frac{1}{n(n-1)}(1 + nx^n)^{1 + \frac{1}{n}} + C \)
• D. \( \frac{1}{n+1}(1 + nx^n)^{1 + \frac{1}{n}} + C \)

Hint:

In problems involving integrals with functions of the form \( \frac{x}{(1 + nx^n)^{1/n}} \), perform substitution to simplify the integral and apply standard power rule integration.

Answer 1

The Correct Option is A

Step 1: Substitute \( f(x) \) into the integral: \[ \int x^{n-2} \cdot \frac{x}{(1 + nx^n)^{1/n}} \, dx = \int \frac{x^{n-1}}{(1 + nx^n)^{1/n}} \, dx \] Step 2: Let \( u = 1 + nx^n \). Then, \( du = n x^{n-1} \, dx \).
Step 3: Substitute into the integral: \[ \int \frac{1}{u^{1/n}} \cdot \frac{du}{n} = \frac{1}{n} \int u^{-1/n} \, du \] Step 4: Integrating \( u^{-1/n} \): \[ \frac{1}{n} \cdot \frac{u^{1 - n}}{1 - n} + C = \frac{1}{n(n-1)} (1 + nx^n)^{1 - \frac{1}{n}} + C \] Thus, the correct answer is: \[ \boxed{\frac{1}{n(n-1)} (1 + nx^n)^{1 - \frac{1}{n}} + C} \]