Question

Evaluate \(\rm \int\frac{3ax}{b^2+c^2x^2}\ dx\).
1. \(\rm \frac{3a}{2c^2}\log\left|{b^2+c^2x^2}\right|+C\)
2. \(\rm \frac{3a}{c^2}\log\left|{b^2+c^2x^2}\right|+C\)
3. \(\rm \frac{3a}{2b^2}\log\left|{b^2+c^2x^2}\right|+C\)
4. None of these.

Answer 1

Correct Answer - Option 1 : \(\rm \frac{3a}{2c^2}\log\left|{b^2+c^2x^2}\right|+C\)

Concept:

• Integration by substitution: If we substitute x = f(t), then dx = f'(t) dt and ∫ f(x) dx = ∫ f[f(t)] f'(t) dt.
• \(\rm \int\frac{dx}{x}=\log x+C\).

 

Calculation:

Let I = \(\rm \int\frac{3ax}{b^2+c^2x^2}\ dx\).

Substituting b² + c²x² = t

⇒ (2c²x)dx = dt

⇒ xdx = \(\rm \frac{dt}{2c^2}\)

⇒ I = \(\rm \frac{3a}{2c^2}\int \frac1t\ dt\)

⇒ I = \(\rm \frac{3a}{2c^2}\log |t| + C\)

⇒ I = \(\rm \frac{3a}{2c^2}\log\left|{b^2+c^2x^2}\right|+C\).