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

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

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integral of sec(x)-tan(x)

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

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

Solution steps

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

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

\int tan (x) d x = - ln ∣ cos (x) ∣

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

Simplify

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

Add a constant to the solution

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

\frac{8.2}{4.2 x ^{2} + 4 x + 5}

\frac{s ^{2}}{( s ^{2} - 1 ) ^{2}}

\frac{10}{x ^{2}}

n = 0 \sum \infty \frac{6}{n ( n + 3 )}

f (t) = \frac{1}{t ^{2}}

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