Question

$\int \sec(x) dx$
Evaluate the integral above​

Answer 1

$&#128075$ Hello ! ☺️

Step-by-step explanation:

∫sec(x)dx =

∫ $\frac{sec(x).(secx + tanx)}{secx + tanx}dx$

∫ $\frac{sec {}^{2} (x) + secxtanx}{secx + tanx}dx$

u = secx + tanx

$du = secx tanx + sec {}^{2}x \: dx$

$∫\frac{1}{u}du$

∫sec(x)dx= ln |u|

$\boxed{\color{gold}{∫sec(x)dx= ln |secx +tanx| + C}}$

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Answer 2

Answer:

$= \ln( | \sec(x) + \tan(x) | ) + C$

Step-by-step explanation:

$\int \sec(x) dx$

multiply and divide by sec x + tan x

$= \int \frac{ \sec(x) ( \sec(x) + \tan(x) ) }{ \sec(x) + \tan(x) } dx$

let u = sec x + tan x

du = (sec x)(sec x + tan x) dx

$= \int \frac{1}{u} du$

$= \ln( |u| ) + C$

$= \boxed{\color{green}{ \ln( | \sec(x) + \tan(x) | ) + C}}$