Question

The integral of \( \int \frac{dx}{x^2[1 + x^4]^{3/4}} \) is:

• A. \( 4(x^{1/4} + 1)^{1/4} + C \)
• B. \( 4(x^{1/4} + 1)^{1/4} + C \)
• C. \( 4(x^4 + 1)^{1/4} + C \)
• D. None of these

Hint:

When dealing with complicated integrals, try using substitution to simplify the expression before attempting further steps.

Answer 1

The Correct Option is D

We are asked to evaluate the integral: \[ \int \frac{dx}{x^2[1 + x^4]^{3/4}} \] Step 1: Use substitution Let \( u = x^4 + 1 \). Differentiating both sides with respect to \( x \): \[ du = 4x^3 \, dx \] Substitute into the integral and simplify it: \[ \int \frac{dx}{x^2 [1 + x^4]^{3/4}} = \int \frac{du}{x^3 [u]^{3/4}} \] However, solving this integral and simplifying further results in an expression that does not match any of the given options. Thus, the correct answer is "None of these."