NCERT Class 12 · Ch 7 Exercise 7.5 · Q19
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 19
Evaluate \(\int \frac{2 x}{\left(x^{2} + 1\right) \left(x^{2} + 3\right)}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 3 \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Polynomial division/partial fractions give \(- \frac{x}{x^{2} + 3} + \frac{x}{x^{2} + 1}\).
- Integrate each rational term using logarithms and ∫dx/(x²+A²)=tan⁻¹(x/A)/A.
- An antiderivative is F(x)=\(\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 3 \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{2 x}{\left(x^{2} + 1\right) \left(x^{2} + 3\right)}\). Add an arbitrary constant C on each domain component.
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