Final answer:
The integral of x/(x⁴+16) dx simplifies to ½ ln(x² + 16) + C by using the substitution u = x² + 16.
Step-by-step explanation:
To evaluate the integral ∫ x/(x⁴+16) dx, we can look for simplifying strategies such as substitution or partial fractions. However, in this case, the integral simplifies directly with a substitution due to the derivative of the denominator being present in the numerator. Specifically, let u = x² + 16, then du = 2x dx. The integral can thus be rewritten as ½ ∫ du/u, which is a standard form and evaluates to ½ ln|u| + C. Substituting back the original expressions for u, we obtain the final result ½ ln(x² + 16) + C, where C is the constant of integration.