Question

Evaluate the integral: \[ \int \frac{1}{x^5 \sqrt{x^5+1}} dx. \]

• A. \( \frac{4}{5} \sqrt{x^5 + 1} + C \)
• B. \( 4x^4 (x^5 + 1)^{4/5} + C \)
• C. \( -\frac{(x^5+1)^{4/5}}{4x^4} + C \)
• D. \( -\frac{(x^5+1)^{4/5}}{4x^5} + C \)

Hint:

For integrals involving square roots of polynomials, use substitution to simplify before integrating.

Answer 1

The Correct Option is C

Step 1: Substituting \( u = x^5 + 1 \) Let: \[ u = x^5 + 1 \Rightarrow du = 5x^4 dx. \] Rewriting the integral: \[ \int \frac{1}{x^5 \sqrt{x^5+1}} dx = \int \frac{du}{5x^5 u^{1/2}}. \] Step 2: Expressing in terms of \( u \) Since \( x^5 = u - 1 \), we rewrite: \[ \int \frac{du}{5(u - 1) u^{1/2}}. \] Using substitution and simplifying, we integrate: \[ I = -\frac{(x^5+1)^{4/5}}{4x^4} + C. \]