1. Partial Fraction Decomposition: Set up the equation: \[\frac{x^2 + 1}{(x^2 + 2)(x^2 + 4)} = \frac{A}{x^2 + 2} + \frac{B}{x^2 + 4}.\]Multiply by the common denominator: \[x^2 + 1 = A(x^2 + 4) + B(x^2 + 2).\]2. Expand and Equate Coefficients: Expand the right side: \[x^2 + 1 = A x^2 + 4A + B x^2 + 2B = (A + B)x^2 + (4A + 2B).\]Equate coefficients for \( x^2 \) and constant terms: \[A + B = 1, \quad 4A + 2B = 1.\]Solve the system for \( A \) and \( B \): From \( A + B = 1 \), we get \( B = 1 - A \). Substitute into the second equation: \[4A + 2(1 - A) = 1 \quad \Rightarrow \quad 4A + 2 - 2A = 1 \quad \Rightarrow \quad 2A = -1 \quad \Rightarrow \quad A = -\frac{1}{2}.\]Substitute \( A \) back into \( B = 1 - A \): \[-\frac{1}{2} + B = 1 \quad \Rightarrow \quad B = \frac{3}{2}.\]3. Rewrite the Integral: Substitute the partial fraction coefficients: \[\int \frac{x^2 + 1}{(x^2 + 2)(x^2 + 4)} \, dx = \int \frac{-\frac{1}{2}}{x^2 + 2} \, dx + \int \frac{\frac{3}{2}}{x^2 + 4} \, dx.\]4. Integrate: Perform the integration for each term: \[\int \frac{-\frac{1}{2}}{x^2 + 2} \, dx = -\frac{1}{2} \cdot \frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right),\]\[\int \frac{\frac{3}{2}}{x^2 + 4} \, dx = \frac{3}{2} \cdot \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right).\]Combine the results: \[\int \frac{x^2 + 1}{(x^2 + 2)(x^2 + 4)} \, dx = -\frac{1}{2\sqrt{2}} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \frac{3}{4} \tan^{-1}\left(\frac{x}{2}\right) + C.\] Final Answer:\[-\frac{1}{2\sqrt{2}} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \frac{3}{4} \tan^{-1}\left(\frac{x}{2}\right) + C\]