$\int e^{x}\sec x\,(1+\tan x)\,dx$ equals
A$e^{x}\cos x+C$
B$e^{x}\sec x+C$
C$e^{x}\sin x+C$
D$e^{x}\tan x+C$
Answer & Solution
Correct answer: B. $e^{x}\sec x+C$
1. Expand: $e^{x}(\sec x+\sec x\tan x)$.
2. Let $f(x)=\sec x$, then $f'(x)=\sec x\tan x$.
3. So the integrand is $e^{x}[f(x)+f'(x)]$, giving $e^{x}f(x)+C$.
4. Result: $e^{x}\sec x+C$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.329_
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