NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q21
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 21
Evaluate \(\int \frac{x^{2} + x + 1}{\left(x + 1\right)^{2} \left(x + 2\right)}\, dx\).
💡 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: \(- 2 \log{\left(\left|{x + 1}\right| \right)} + 3 \log{\left(\left|{x + 2}\right| \right)} - \frac{1}{x + 1}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Partial fractions give \(\frac{3}{x + 2} - \frac{2}{x + 1} + \frac{1}{\left(x + 1\right)^{2}}\). Integrate the simple and repeated linear factors.
- An antiderivative is F(x)=\(- 2 \log{\left(\left|{x + 1}\right| \right)} + 3 \log{\left(\left|{x + 2}\right| \right)} - \frac{1}{x + 1}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{2} + x + 1}{\left(x + 1\right)^{2} \left(x + 2\right)}\). Add an arbitrary constant C on each domain component.
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