NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q38
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 38
Evaluate \(\int \frac{1}{e^{x} + e^{- x}}\, dx\).
💡 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: A: \(\operatorname{atan}{\left(e^{x} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Multiply numerator and denominator by eˣ; put u=eˣ and integrate du/(1+u²).
- An antiderivative is F(x)=\(\operatorname{atan}{\left(e^{x} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{e^{x} + e^{- x}}\). Add an arbitrary constant C on each domain component.
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