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

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

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integral of tan(4x)-cot(x/4)

\int tan (4 x) - cot (\frac{x}{4}) d x

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

Solution steps

\int tan (4 x) - cot (\frac{x}{4}) 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 (4 x) d x - \int cot (\frac{x}{4}) d x

\int tan (4 x) d x = - \frac{1}{4} ln ∣ cos (4 x) ∣

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

= - \frac{1}{4} ln ∣ cos (4 x) ∣ - 4 ln sin (\frac{x}{4})

Add a constant to the solution

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

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