What is the antiderivative of $\frac{1}{cos x} (2 + sin x)$ $1/cosx(2+sinx)$ ?

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Answer 1 · 2017-03-05T21:55:01

$\int \frac{sin x + 2}{cos x} d x$ $int frac{sinx+2}{cosx}dx$
$= ln | sec x | + 2 ln | sec x + tan x | + C$ $=ln|secx|+2ln|secx+tanx|+C$

$\int \frac{sin x + 2}{cos x} d x$ $int frac{sinx+2}{cosx}dx$

$\int \frac{sin x}{cos x} + \frac{2}{cos x} d x$ $int frac{sinx}{cosx}+frac{2}{cosx}dx$

$= \int (tan x) d x + 2 \int (sec x) d x$ $=int (tanx)dx + 2int(secx)dx$

$= ln | sec x | + 2 ln | sec x + tan x | + C$ $=ln|secx|+2ln|secx+tanx|+C$

Use logarithm rules:
$= ln | sec x | + {ln (sec x + tan x)}^{2} + C$ $=ln|secx|+ln(secx+tanx)^2+C$

$= ln ∣ ∣ (sec x) {(sec x + tan x)}^{2} ∣ ∣ + C$ $=ln|(secx)(secx + tanx)^2|+C$

What is the antiderivative of 1cosx(2+sinx)1/cosx(2+sinx)?