What is the integral of (2+cot(x))/(cos(x)) ?

The integral of (2+cot(x))/(cos(x)) is 2ln|tan(x)+sec(x)|+ln|tan(x/2)|+C

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integral of (2+cot(x))/(cos(x))

\int \frac{2 + cot ( x )}{cos ( x )} d x

2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2}) + C

Solution steps

\int \frac{2 + cot ( x )}{cos ( x )} d x

Expand

\frac{2 + cot ( x )}{cos ( x )} : \frac{2}{cos ( x )} + \frac{cot ( x )}{cos ( x )}

= \int \frac{2}{cos ( x )} + \frac{cot ( x )}{cos ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2}{cos ( x )} d x + \int \frac{cot ( x )}{cos ( x )} d x

\int \frac{2}{cos ( x )} d x = 2 ln ∣ tan (x) + sec (x) ∣

\int \frac{cot ( x )}{cos ( x )} d x = ln tan (\frac{x}{2})

= 2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2})

Add a constant to the solution

= 2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2}) + C

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What is the integral of (2+cot(x))/(cos(x)) ?
The integral of (2+cot(x))/(cos(x)) is 2ln|tan(x)+sec(x)|+ln|tan(x/2)|+C