NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q9
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 9
Evaluate \(\int \frac{\cos{\left(x \right)}}{\sqrt{4 - \sin^{2}{\left(x \right)}}}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{2} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Put u=sin x, du=cos x dx.
- An antiderivative is F(x)=\(\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{2} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{\cos{\left(x \right)}}{\sqrt{4 - \sin^{2}{\left(x \right)}}}\). Add an arbitrary constant C on each domain component.
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