e^ x x(1+ x) d x equals - Vidyadip

Question

$\int e^{x} \sec x(1+\tan x) d x$ equals

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Answer

$\int e^{x} \sec x(1+\tan x) d x$
Let
$\mathrm{I}=\int e^{x} \sec x(1+\tan x) d x=\int e^{x}(\sec x+\sec x \tan x) d x ...(i)$
Also, let $sec x = f(x)$
$\Rightarrow \sec x \tan x = f’(x)$
We know that $\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+C$
Thus $(i)$ gives, $I = e^{x }sec x + C$

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