NCERT Class 12 · Ch 7 Exercise 7.5 · Q18
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 18
Evaluate \(\int \frac{\left(x^{2} + 1\right) \left(x^{2} + 2\right)}{\left(x^{2} + 3\right) \left(x^{2} + 4\right)}\, dx\).
💡 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: \(x - 3 \operatorname{atan}{\left(\frac{x}{2} \right)} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{3}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Polynomial division/partial fractions give \(1 - \frac{6}{x^{2} + 4} + \frac{2}{x^{2} + 3}\).
- Integrate each rational term using logarithms and ∫dx/(x²+A²)=tan⁻¹(x/A)/A.
- An antiderivative is F(x)=\(x - 3 \operatorname{atan}{\left(\frac{x}{2} \right)} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{3}\).
- Differentiate this expression to check that F′(x)=\(\frac{\left(x^{2} + 1\right) \left(x^{2} + 2\right)}{\left(x^{2} + 3\right) \left(x^{2} + 4\right)}\). Add an arbitrary constant C on each domain component.
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