Question

Evaluate \( \int \frac{dx}{x^2 (x^4 + 1)^{3/4}} \):

• A. \( -(x^4 + 1)^{\frac{1}{4}} + C \)
• B. \( (x^4 + 1)^{\frac{1}{4}} + C \)
• C. \( -\frac{(x^4 + 1)^{1/4}}{x^4} + C \)
• D. \( \frac{(x^4 + 1)}{x^4} + C \)
• E. \( \frac{(x^4 + 1)^{3/4}}{x^4} + C \)

Hint:

Substituting expressions involving powers of polynomials can simplify integration significantly.

Answer 1

The Correct Option is C

We begin by applying substitution. Define: \[ u = x^4 + 1 \] which gives: \[ du = 4x^3 \, dx \] Next, we rewrite the given integral in terms of \( u \).
Noticing the presence of \( x^2 \) and \( x^3 \) in the integral, we express \( x^2 \) as \( x^3 \cdot x^{-1} \), transforming the integral into: \[ \int \frac{dx}{x^2 (x^4 + 1)^{3/4}} = \int \frac{du}{x^2 u^{3/4}} \] Upon further simplification and solving, we obtain: \[ \int \frac{dx}{x^2 (x^4 + 1)^{3/4}} = -\frac{(x^4 + 1)^{1/4}}{x^4} + C \] 
Thus, the correct answer is \( -\frac{(x^4 + 1)^{1/4}}{x^4} + C \).