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

The integral of (tan(2x)-sec(2x)) is -1/2 ln|cos(2x)|-1/2 ln|tan(2x)+sec(2x)|+C

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integral of (tan(2x)-sec(2x))

\int (tan (2 x) - sec (2 x)) d x

- \frac{1}{2} ln ∣ cos (2 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

Solution steps

\int tan (2 x) - sec (2 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 (2 x) d x - \int sec (2 x) d x

\int tan (2 x) d x = - \frac{1}{2} ln ∣ cos (2 x) ∣

\int sec (2 x) d x = \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

= - \frac{1}{2} ln ∣ cos (2 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ cos (2 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

\int \frac{x}{x ^{2} + 4 x + 5} d x

\frac{9 - x ^{2}}{32 x}

f (x) = 3 x^{3} (x^{2} - 2)

f (x) = (3 x^{2} - 2) (5 x + 5)

\int \frac{1}{1 - cos ^{2} ( x )} d x

What is the integral of (tan(2x)-sec(2x)) ?
The integral of (tan(2x)-sec(2x)) is -1/2 ln|cos(2x)|-1/2 ln|tan(2x)+sec(2x)|+C