NCERT Class 12 · Ch 7 Exercise 7.2 · Q33
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 33
Evaluate \(\int \frac{1}{1 - \tan{\left(x \right)}}\, dx\).
💡 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: \(\frac{x}{2} - \frac{\log{\left(\left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right| \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Rewrite cos x/(cos x−sin x) as 1/2+(sin x+cos x)/[2(cos x−sin x)].
- An antiderivative is F(x)=\(\frac{x}{2} - \frac{\log{\left(\left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{1 - \tan{\left(x \right)}}\). Add an arbitrary constant C on each domain component.
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