NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 25

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Let u=sin²x; denominator is 2u²−2u+1 and sin x cos x dx=du/2.
3. An antiderivative is F(x)=\(\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}\).
4. At the upper and lower limits, F(b)=\(0\) and F(a)=\(- \frac{\pi}{8}\).
5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi}{8}\).