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NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q24

NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 24

Evaluate \(\int\limits_{\frac{\pi}{2}}^{\pi} \frac{\left(1 - \sin{\left(x \right)}\right) e^{x}}{1 - \cos{\left(x \right)}}\, dx\).

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(e^{\frac{\pi}{2}}\)
  1. Work on each open interval where the original integrand is real and defined.
  2. Recognize the derivative of −eˣcot(x/2).
  3. An antiderivative is F(x)=\(- e^{x} \cot{\left(\frac{x}{2} \right)}\).
  4. At the upper and lower limits, F(b)=\(0\) and F(a)=\(- e^{\frac{\pi}{2}}\).
  5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(e^{\frac{\pi}{2}}\).

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