NCERT Class 12 · Ch 7 Exercise 7.2 · Q25
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 25
Evaluate \(\int \frac{1}{\left(1 - \tan{\left(x \right)}\right)^{2} \cos^{2}{\left(x \right)}}\, dx\).
💡 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: \(\frac{1}{1 - \tan{\left(x \right)}}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=1−tan x, du=−sec²x dx; integrate −u^(−2)du.
- An antiderivative is F(x)=\(\frac{1}{1 - \tan{\left(x \right)}}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\left(1 - \tan{\left(x \right)}\right)^{2} \cos^{2}{\left(x \right)}}\). Add an arbitrary constant C on each domain component.
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