We are given the integral $I = \int \frac{3ax}{b^2 + c^2x^2} dx$. We observe that the derivative of the denominator $b^2 + c^2x^2$ is $2c^2x$.
Let $u = b^2 + c^2x^2$. Then, the differential $du = 2c^2x \, dx$, which implies $x \, dx = \frac{du}{2c^2}$.
Substituting these into the integral: $$I = \int \frac{3a}{u} \cdot \frac{du}{2c^2} = \frac{3a}{2c^2} \int \frac{1}{u} du$$ $$I = \frac{3a}{2c^2} \log |u| + K = \frac{3a}{2c^2} \log |b^2 + c^2x^2| + K$$
Comparing $\frac{3a}{2c^2} \log |b^2 + c^2x^2| + K$ with $A \log |b^2 + c^2x^2| + K$, we find that $A = \frac{3a}{2c^2}$.
Final Answer: $\frac{3a}{2c^2}$
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