NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 27

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Let u=sin x−cos x, du=(sin x+cos x)dx and sin 2x=1−u².
3. An antiderivative is F(x)=\(\operatorname{asin}{\left(\sin{\left(x \right)} - \cos{\left(x \right)} \right)}\).
4. At the upper and lower limits, F(b)=\(- \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\) and F(a)=\(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\).
5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(- 2 \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\).