1 markMCQSection A

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

Q: 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

**Correct option: (D) $\frac{3a}{2c^2}$** Substituting $u = b^2+c^2x^2 \text{ gives } du = 2c^2x\,dx$. The integral becomes $\left(\frac{3a}{2c^2}\right)\int \frac{du}{u} = \left(\frac{3a}{2c^2}\right)\log|u|+K$, so A = $\frac{3a}{2c^2}$.

Integrals

Integration by Substitution

Options

(A)

3a

(B)

\frac{3a}{2b^2}

(C)

\frac{3a}{b^2 c^2}

(D)

\frac{3a}{2c^2}

Official Answer

Correct option: (D) $\frac{3a}{2c^2}$

Substituting $u = b^2+c^2x^2 \text{ gives } du = 2c^2x\,dx$ . The integral becomes $\left(\frac{3a}{2c^2}\right)\int \frac{du}{u} = \left(\frac{3a}{2c^2}\right)\log|u|+K$ , so A = $\frac{3a}{2c^2}$ .

integration by substitutionlog integraldu=2c^2x dxindefinite integralconstant of integrationx over quadratic integral

Marking Scheme

11 mark: correct option (D) selected — requires correctly identifying the substitution and its derivative factor.

Hint

Substitute $u = b^2 + c^2x^2 \text{ so that } du = 2c^2x\,dx$ , then rewrite the integral purely in terms of u.

Quick Oral Answer

Using $u = b^2+c^2x^2,\ du = 2c^2x\,dx$ , the integral reduces to $\frac{3a}{2c^2}\log|u|, \text{ so } A = \frac{3a}{2c^2}$ , option (D).

Analysis & Explanation

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.

Common Mistakes

1Forgetting to divide by the derivative factor 2c² when substituting $u = b^2+c^2x^2$ , leading to answer (A).
2Mixing up which constant (b or c) belongs in the denominator, leading to answer (B) or (C).
3Not simplifying $3a \times \left(\frac{1}{2c^2}\right)$ correctly and leaving an incorrect combined fraction.

Interesting Facts

This substitution pattern — recognizing that the numerator is (a constant times) the derivative of the denominator's variable part — is one of the most frequently tested integration techniques in CBSE board exams.

Integrals of the form $\int \frac{f'(x)}{f(x)}\,dx$ always reduce to $\log|f(x)|+K$ , a shortcut that avoids lengthy substitution steps once the pattern is recognized.

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Frequently Asked Questions

How do you recognize when to use log-form integration?

Whenever the integrand looks like a constant multiple of $f'(x)/f(x)$ , i.e., the numerator resembles the derivative of the denominator (up to a constant factor), substituting $u = f(x)$ will reduce the integral to $\int \frac{du}{u} = \log|u|+K$ .

What is the role of the constant K in this problem?

K is the arbitrary constant of integration, required for any indefinite integral since differentiating a constant gives zero — infinitely many antiderivatives differ only by this constant.