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

The integral of (tan(x)+sec(x)) is -ln|cos(x)|+ln|tan(x)+sec(x)|+C

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

\int (tan (x) + sec (x)) d x

- ln ∣ cos (x) ∣ + ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

\int tan (x) + sec (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 tan (x) d x + \int sec (x) d x

\int tan (x) d x = - ln ∣ cos (x) ∣

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

= - ln ∣ cos (x) ∣ + ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) ∣ + ln ∣ tan (x) + sec (x) ∣ + C

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