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

The integral of tan(x)-2cot(x) is -ln|cos(x)|-2ln|sin(x)|+C

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

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

- ln ∣ cos (x) ∣ - 2 ln ∣ sin (x) ∣ + C

Solution steps

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

\int tan (x) d x = - ln ∣ cos (x) ∣

\int 2 cot (x) d x = 2 ln ∣ sin (x) ∣

= - ln ∣ cos (x) ∣ - 2 ln ∣ sin (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) ∣ - 2 ln ∣ sin (x) ∣ + C

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What is the integral of tan(x)-2cot(x) ?
The integral of tan(x)-2cot(x) is -ln|cos(x)|-2ln|sin(x)|+C