NCERT Class 12 · Ch 7 Exercise 7.3 · Q8
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 8
Evaluate \(\int \frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx\).
💡 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: \(- x + 2 \tan{\left(\frac{x}{2} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- The fraction is tan²(x/2)=sec²(x/2)−1.
- An antiderivative is F(x)=\(- x + 2 \tan{\left(\frac{x}{2} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\). Add an arbitrary constant C on each domain component.
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