What is the integral of 3-tan(x/6) ?

The integral of 3-tan(x/6) is 3x+6ln|cos(x/6)|+C

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integral of 3-tan(x/6)

\int 3 - tan (\frac{x}{6}) d x

3 x + 6 ln cos (\frac{x}{6}) + C

Solution steps

\int 3 - tan (\frac{x}{6}) 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 3 d x - \int tan (\frac{x}{6}) d x

\int 3 d x = 3 x

\int tan (\frac{x}{6}) d x = - 6 ln cos (\frac{x}{6})

= 3 x - (- 6 ln cos (\frac{x}{6}))

Simplify

= 3 x + 6 ln cos (\frac{x}{6})

Add a constant to the solution

= 3 x + 6 ln cos (\frac{x}{6}) + C

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What is the integral of 3-tan(x/6) ?
The integral of 3-tan(x/6) is 3x+6ln|cos(x/6)|+C