NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q32
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 32
Evaluate \(\int\limits_{1}^{3} \frac{1}{x^{2} \left(x + 1\right)}\, dx\).
💡 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: \(\log{\left(\frac{2}{3} \right)} + \frac{2}{3}\)
- Work on each open interval where the original integrand is real and defined.
- Decompose as −1/x+1/x²+1/(x+1).
- An antiderivative is F(x)=\(- \log{\left(\left|{x}\right| \right)} + \log{\left(\left|{x + 1}\right| \right)} - \frac{1}{x}\).
- At the upper and lower limits, F(b)=\(- \log{\left(3 \right)} - \frac{1}{3} + \log{\left(4 \right)}\) and F(a)=\(-1 + \log{\left(2 \right)}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\log{\left(\frac{2}{3} \right)} + \frac{2}{3}\).
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