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 secx = f(x)
$\Rightarrow$ secxtanx = 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, $I = e^xsec\ x + C$

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