NCERT Class 12 · Ch 7 Exercise 7.6 · Q12
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 12
Evaluate \(\int x \sec^{2}{\left(x \right)}\, dx\).
💡 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: \(x \tan{\left(x \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Choose u=\(x\) and v=\(\tan{\left(x \right)}\), so dv=\(\tan^{2}{\left(x \right)} + 1\)dx and du=\(1\)dx.
- Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(\tan{\left(x \right)}\). Integrate it and subtract from uv.
- An antiderivative is F(x)=\(x \tan{\left(x \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(x \sec^{2}{\left(x \right)}\). Add an arbitrary constant C on each domain component.
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