NCERT Class 12 · Ch 7 Exercise 7.3 · Q19
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 19
Evaluate \(\int \frac{1}{\sin{\left(x \right)} \cos^{3}{\left(x \right)}}\, dx\).
💡 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: \(\log{\left(\left|{\tan{\left(x \right)}}\right| \right)} + \frac{\tan^{2}{\left(x \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Let u=tan x; the integral becomes ∫(u+1/u)du.
- An antiderivative is F(x)=\(\log{\left(\left|{\tan{\left(x \right)}}\right| \right)} + \frac{\tan^{2}{\left(x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sin{\left(x \right)} \cos^{3}{\left(x \right)}}\). Add an arbitrary constant C on each domain component.
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