NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 30

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Put u=sin x; sin 2x dx=2u du. Integrate 2u tan⁻¹u by parts.
3. An antiderivative is F(x)=\(\left(\sin^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)}\).
4. At the upper and lower limits, F(b)=\(-1 + \frac{\pi}{2}\) and F(a)=\(0\).
5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(-1 + \frac{\pi}{2}\).