Question:
Evaluate the following integrals:
$\int e^{x} \sec x(1+\tan x) d x$
Solution:
Let $I=\int e^{x} \sec x(1+\tan x) d x$
$=\int e^{x} \operatorname{secx} d x+\int e^{x} \operatorname{secxtan} x d x$
Integrating by parts,
$=e^{x} \operatorname{secxdx}-\int e^{x} \frac{d}{d x} \operatorname{secxdx}+\int e^{x} \operatorname{secxtan} x d x$
$=e^{x} \operatorname{secxdx}-\int e^{x} \operatorname{secxtan} x d x+\int e^{x} \operatorname{secxtan} x d x$
$=e^{x} \operatorname{secx} d x+c$
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