NCERT Class 12 · Ch 7 Exercise 7.8 · Q8
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 8
Evaluate \(\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc{\left(x \right)}\, dx\).
💡 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: \(\log{\left(\frac{-1 + \sqrt{2}}{2 - \sqrt{3}} \right)}\)
- Work on each open interval where the original integrand is real and defined.
- Use linearity and the elementary power, logarithm, trigonometric and exponential rules. Where a product x eˣ occurs use integration by parts; for rational terms split off the denominator derivative.
- An antiderivative is F(x)=\(\log{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}\).
- At the upper and lower limits, F(b)=\(\log{\left(-1 + \sqrt{2} \right)}\) and F(a)=\(\log{\left(2 - \sqrt{3} \right)}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\log{\left(\frac{-1 + \sqrt{2}}{2 - \sqrt{3}} \right)}\).
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