NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q14
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 14
Evaluate \(\int \frac{1}{\left(x^{2} + 1\right) \left(x^{2} + 4\right)}\, dx\).
💡 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: \(- \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{6} + \frac{\operatorname{atan}{\left(x \right)}}{3}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Partial fractions: [1/(x²+1)−1/(x²+4)]/3.
- An antiderivative is F(x)=\(- \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{6} + \frac{\operatorname{atan}{\left(x \right)}}{3}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\left(x^{2} + 1\right) \left(x^{2} + 4\right)}\). Add an arbitrary constant C on each domain component.
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