If $\int \frac{3ax}{b^2 + c^2 x^2}\,dx = A \log |b^2 + c^2 x^2| + K$, then the value of A is

This tests the standard substitution technique for integrals of the form $\int \frac{x}{\text{quadratic in } x^2}\,dx$.

Setting up the substitution

• Let $u = b^2 + c^2x^2$. Then $\frac{du}{dx} = 2c^2x, \text{ so } x\,dx = \frac{du}{2c^2}$.

Substituting into the integral

• $\int \frac{3ax}{b^2 + c^2x^2}\,dx = 3a\int \frac{x\,dx}{b^2+c^2x^2} = 3a\int \frac{du/(2c^2)}{u} = \frac{3a}{2c^2}\int \frac{du}{u}$.

Integrating

• $\frac{3a}{2c^2}\int \frac{du}{u} = \frac{3a}{2c^2}\log|u| + K = \frac{3a}{2c^2}\log|b^2+c^2x^2| + K$.

Why (D) is correct

• Comparing with $A\log|b^2+c^2x^2|+K$, we get $A = \frac{3a}{2c^2}$, matching option (D).

Why the other options are wrong

• (A) 3a ignores the substitution factor entirely, as if du = x dx directly.
• (B) $\frac{3a}{2b^2}$ mistakenly uses $b^2$ instead of $c^2$ in the denominator, confusing the two constants.
• (C) $\frac{3a}{b^2c^2}$ incorrectly combines both constants in the denominator and drops the factor of 2.