NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q11
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 11
Evaluate \(\int \frac{1}{\cos{\left(a + x \right)} \cos{\left(b + x \right)}}\, dx\).
💡 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: For sin(a−b)≠0: log|cos(x+b)/cos(x+a)|/sin(a−b)+C. If a−b=kπ, k∈ℤ: (−1)^k tan(x+a)+C. Exclude the original cosine zeros.
- Work on each open interval where the original integrand is real and defined.
- Use tan(x+a)−tan(x+b)=sin(a−b)/[cos(x+a)cos(x+b)]. If a−b=kπ, use (−1)ᵏtan(x+a).
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{\frac{\cos{\left(b + x \right)}}{\cos{\left(a + x \right)}}}\right| \right)}}{\sin{\left(a - b \right)}}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\cos{\left(a + x \right)} \cos{\left(b + x \right)}}\). Add an arbitrary constant C on each domain component.
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