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

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

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

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

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

t an g e n t f (x) 3 x^{2} - 2 x + 8, at x = - 1

d er i v a t i v e \frac{x ^{3} + 9}{x ^{3}}

\frac{d}{d x} (\frac{5 x ^{4} + x ^{2}}{2})

d er i v a t i v e 2 sec^{2} (x) tan (x)

d er i v a t i v e cot (x^{2} + π x)