NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 18

✅ Step-by-Step Solution

1. Work where the radicand is positive and sin α≠0.
2. Put u=cot x. Since sin(x+α)=sin x(cos α+u sin α), the square root is sin²x√(cos α+u sin α).
3. As du=−cosec²x dx, the integral becomes −∫du/√(cos α+u sin α).
4. An antiderivative is F(x)=\(- \frac{2 \sqrt{\sin{\left(\alpha \right)} \cot{\left(x \right)} + \cos{\left(\alpha \right)}}}{\sin{\left(\alpha \right)}}\).
5. Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt{\sin^{3}{\left(x \right)} \sin{\left(\alpha + x \right)}}}\). Add an arbitrary constant C on each domain component.
6. If sin α=0 and cos α=1 the real integrand is cosec²x and the answer is −cot x+C. If cos α=−1, no nonzero real radicand is positive.