Question

Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]

Hint:

When dealing with integrals involving logarithmic functions, use integration by parts or substitution to break the integral into manageable parts.

Answer 1

First, observe that: \[ \frac{\sqrt{x^2 + 1}}{x^2} = \frac{1}{x} \cdot \frac{\sqrt{x^2 + 1}}{x} \] Thus, we can rewrite the integral as: \[ J = \int \left( \frac{\sqrt{x^2 + 1}}{x} \log(x^2 + 1) - 2 \cdot \frac{\sqrt{x^2 + 1}}{x} \log x \right) \, dx \] Now, break it into two parts: \[ J_1 = \int \frac{\sqrt{x^2 + 1}}{x} \log(x^2 + 1) \, dx \] \[ J_2 = \int -2 \cdot \frac{\sqrt{x^2 + 1}}{x} \log x \, dx \] Each of these integrals can be solved using integration by parts or substitution. The final result for both integrals can be obtained as: \[ J = J_1 + J_2 \]