What is the integral of (sec(x)tan(x))/(2+sec(x)) ?

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

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integral of (sec(x)tan(x))/(2+sec(x))

\int \frac{sec ( x ) tan ( x )}{2 + sec ( x )} d x

ln ∣ 2 + sec (x) ∣ + C

Solution steps

\int \frac{sec ( x ) tan ( x )}{2 + sec ( x )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 2 + sec (x)

= ln ∣ 2 + sec (x) ∣

Add a constant to the solution

= ln ∣ 2 + sec (x) ∣ + C

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