Discussion
\(\int e^{x}\sec x(1 + tanx)\,dx\)
Answer: C
The integral is \(e^{x}\sec x + C\).
Expand the integrand as \(e^{x}(\sec x + \sec x\tan x)\). This has the form \(e^{x}\left(f(x) + f'(x)\right)\) with \(f(x) = \sec x\), since \(\frac{d}{dx}\sec x = \sec x\tan x\), and
\[ \int e^{x}\left(f(x) + f'(x)\right)dx = e^{x}f(x) + C \]
Check: \(\frac{d}{dx}(e^{x}\sec x) = e^{x}\sec x + e^{x}\sec x\tan x\). \(e^{x}\tan x\) would need \(\tan x + \sec^{2}x\) in the bracket, and \(e^{x}\sec x\tan x\) differentiates to something with \(\sec^{3}x\).
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