NCERT Class 12 · Ch 7 Exercise 7.3 · Q15
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 15
Evaluate \(\int \tan^{3}{\left(2 x \right)} \sec{\left(2 x \right)}\, dx\).
💡 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: \(\frac{\sec^{3}{\left(2 x \right)}}{6} - \frac{\sec{\left(2 x \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Write tan³u sec u=tan u sec u(sec²u−1); put v=sec u, du=2dx.
- An antiderivative is F(x)=\(\frac{\sec^{3}{\left(2 x \right)}}{6} - \frac{\sec{\left(2 x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\tan^{3}{\left(2 x \right)} \sec{\left(2 x \right)}\). Add an arbitrary constant C on each domain component.
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