What is the integral of tan(1/3 x) ?

The integral of tan(1/3 x) is -3ln|cos(x/3)|+C

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integral of tan(1/3 x)

\int tan (\frac{1}{3} x) d x

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

Solution steps

\int tan (\frac{1}{3} x) d x

Rewrite using trig identities

= \int \frac{sin ( \frac{x}{3} )}{cos ( \frac{x}{3} )} d x

Apply u-substitution

: \int - \frac{3}{u} d u

= \int - \frac{3}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 3 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 3 ln ∣ u ∣

Substitute back

u = cos (\frac{x}{3})

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

Add a constant to the solution

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

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What is the integral of tan(1/3 x) ?
The integral of tan(1/3 x) is -3ln|cos(x/3)|+C