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# Choose the correct answer in $\int e^x\sec x(1+\tan x)\;dx$ equals

Toolbox:
• (i) $\int e^x[f(x)+f'(x)]dx=e^x(f(x))+c$
• (ii) $\frac{d}{dx}(\sec x)=\sec x \tan x$
Given $I=\int e^x \sec x(1+\tan x)dx$

$\qquad=\int e^x [\sec x+\sec x tan x]dx$

Let $f(x)=\sec x$

On differentiating with respect to x,

$f'(x)=\sec x \tan x$

Therefore The integral function of the form

$\int e^x[f(x)+f'(x)]=e^x(f(x))+c$

Therefore $\int e^x[\sec x+\sec x \tan x ]dx=e^x \sec x +c$

Hence the correct answer is B