What is the integral of tan^2(x/3) ?

The integral of tan^2(x/3) is -x+3tan(x/3)+C

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integral of tan^2(x/3)

\int tan^{2} (\frac{x}{3}) d x

- x + 3 tan (\frac{x}{3}) + C

Solution steps

\int tan^{2} (\frac{x}{3}) d x

Rewrite using trig identities

= \int - 1 + sec^{2} (\frac{x}{3}) 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 1 d x + \int sec^{2} (\frac{x}{3}) d x

\int 1 d x = x

\int sec^{2} (\frac{x}{3}) d x = 3 tan (\frac{x}{3})

= - x + 3 tan (\frac{x}{3})

Add a constant to the solution

= - x + 3 tan (\frac{x}{3}) + C

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What is the integral of tan^2(x/3) ?
The integral of tan^2(x/3) is -x+3tan(x/3)+C