NCERT Class 12 · Ch 7 Exercise 7.2 · Q29
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 29
Evaluate \(\int \log{\left(\sin{\left(x \right)} \right)} \cot{\left(x \right)}\, dx\).
💡 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: \(\frac{\log{\left(\left|{\sin{\left(x \right)}}\right| \right)}^{2}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=log(sin x), du=cot x dx; require sin x>0.
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{\sin{\left(x \right)}}\right| \right)}^{2}}{2}\).
- Differentiate this expression to check that F′(x)=\(\log{\left(\sin{\left(x \right)} \right)} \cot{\left(x \right)}\). Add an arbitrary constant C on each domain component.
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