What is the integral of (sec(θ)tan(θ))/(1+sec(θ)) ?

The integral of (sec(θ)tan(θ))/(1+sec(θ)) is ln|1+sec(θ)|+C

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integral of (sec(θ)tan(θ))/(1+sec(θ))

\int \frac{sec ( θ ) tan ( θ )}{1 + sec ( θ )} d θ

ln ∣ 1 + sec (θ) ∣ + C

Solution steps

\int \frac{sec ( θ ) tan ( θ )}{1 + sec ( θ )} d θ

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 = 1 + sec (θ)

= ln ∣ 1 + sec (θ) ∣

Add a constant to the solution

= ln ∣ 1 + sec (θ) ∣ + C

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What is the integral of (sec(θ)tan(θ))/(1+sec(θ)) ?
The integral of (sec(θ)tan(θ))/(1+sec(θ)) is ln|1+sec(θ)|+C