What is the integral of sec(4x)-cot(4x) ?

The integral of sec(4x)-cot(4x) is 1/4 ln|tan(4x)+sec(4x)|-1/4 ln|sin(4x)|+C

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integral of sec(4x)-cot(4x)

\int sec (4 x) - cot (4 x) d x

\frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{1}{4} ln ∣ sin (4 x) ∣ + C

Solution steps

\int sec (4 x) - cot (4 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 (4 x) d x - \int cot (4 x) d x

\int sec (4 x) d x = \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣

\int cot (4 x) d x = \frac{1}{4} ln ∣ sin (4 x) ∣

= \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{1}{4} ln ∣ sin (4 x) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{1}{4} ln ∣ sin (4 x) ∣ + C

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What is the integral of sec(4x)-cot(4x) ?
The integral of sec(4x)-cot(4x) is 1/4 ln|tan(4x)+sec(4x)|-1/4 ln|sin(4x)|+C