Exercise 7.4 - Question 8
Problem
$$\int \frac{x^2}{\sqrt{x^6 + a^6}} \, dx$$
Step-by-Step Solution
Step 1: Recognize the Pattern
Notice that x⁶ + a⁶ = (x³)² + (a³)² and the derivative of x³ is 3x².
Step 2: Substitute
Let u = x³, then du = 3x² dx, so x² dx = du/3:
$$\int \frac{x^2}{\sqrt{x^6 + a^6}} \, dx = \frac{1}{3}\int
\frac{du}{\sqrt{u^2 + a^6}}$$
Step 3: Apply Standard Formula
$$\int \frac{1}{\sqrt{u^2 + b^2}} \, du = \ln\left|u + \sqrt{u^2 +
b^2}\right| + C$$
With b = a³:
$$= \frac{1}{3}\ln\left|u + \sqrt{u^2 + a^6}\right| + C$$
Step 4: Substitute Back
$$= \frac{1}{3}\ln\left|x^3 + \sqrt{x^6 + a^6}\right| + C$$
✅ Final Answer
$$\int \frac{x^2}{\sqrt{x^6 + a^6}} \, dx = \frac{1}{3}\ln\left|x^3 +
\sqrt{x^6 + a^6}\right| + C$$