NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q36
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 36
Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} 2 \tan^{3}{\left(x \right)}\, dx\).
💡 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: \(1 - \log{\left(2 \right)}\)
- Work on each open interval where the original integrand is real and defined.
- Use tan³x=tan x sec²x−tan x.
- An antiderivative is F(x)=\(2 \log{\left(\left|{\cos{\left(x \right)}}\right| \right)} + \tan^{2}{\left(x \right)}\).
- At the upper and lower limits, F(b)=\(2 \log{\left(\frac{\sqrt{2}}{2} \right)} + 1\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(1 - \log{\left(2 \right)}\).
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