Integrate ∫ x^2/(√ x^6 +16)dx

Answer 1

Let x³ = 4 tanθ such that 3x²dx = 4 sec²θdθ now x²dx = 4sec²θdθ/3 the integral becomes 4/3∫sec²θdθ/4secθ. This gives ∫secθdθ/3. Doing the integration gives 1/3( ln| secθ + tanθ|). Using the triangle to relate the angles to x where x³ is the vertical, 4 is the horizontal and the hypotenuse is √(x⁶ +16) the final answer is l/3ln| (√(x⁶ +16) + x³)/4|.