Evaluate the following integrals:

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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