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

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

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integral of tan(-x/3)

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

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

Solution steps

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

Simplify

tan (- \frac{x}{3}) : - tan (\frac{x}{3})

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

Take the constant out:

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

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

Simplify

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