NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 29

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Put u=sin x−cos x. The denominator becomes 25−16u²; use partial fractions.
3. An antiderivative is F(x)=\(\frac{\log{\left(\left|{\frac{4 \sin{\left(x \right)} - 4 \cos{\left(x \right)} + 5}{- 4 \sin{\left(x \right)} + 4 \cos{\left(x \right)} + 5}}\right| \right)}}{40}\).
4. At the upper and lower limits, F(b)=\(0\) and F(a)=\(- \frac{\log{\left(9 \right)}}{40}\).
5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\log{\left(3 \right)}}{20}\).