NCERT Class 12 · Ch 7 Exercise 7.9 · Q7
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 7
Evaluate \(\int\limits_{-1}^{1} \frac{1}{x^{2} + 2 x + 5}\, dx\).
💡 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: \(\frac{\pi}{8}\)
- Work on each open interval where the original integrand is real and defined.
- Complete the square: (x+1)²+4; set u=(x+1)/2.
- An antiderivative is F(x)=\(\frac{\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}}{2}\).
- At the upper and lower limits, F(b)=\(\frac{\pi}{8}\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi}{8}\).
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