Class 12 Integrals · Short Answer2 Marksproduct_derivative
Class 12 Maths · Integrals · Short Answer #138
Find one function f satisfying \(\int e^x(\tan x+1)\sec x\,dx=e^x f(x)+C\) on an interval with cos x≠0.
✅ Step-by-Step Solution
Final Answer: One choice is f(x)=sec x; the full family is sec x+K e^(−x).
- Differentiate e^x sec x: its derivative is e^x(sec x+sec x tan x).
- This is precisely the integrand, so e^x f=e^x sec x+K for a constant K.
- A valid choice is f=sec x; generally f=sec x+K e^(−x).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Differentiate e^x sec x: its derivative is e^x(sec x+sec x tan x).
- [1 mark] This is precisely the integrand, so e^x f=e^x sec x+K for a constant K.
- [0.5 marks] A valid choice is f=sec x; generally f=sec x+K e^(−x).
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