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

The integral of cotan(x/3) is -3coln|cos(x/3)|+C

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integral of cotan(x/3)

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

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

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= co (- 3 ln ∣ u ∣)

Substitute back

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

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

Simplify

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

Add a constant to the solution

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

y^{^{''}} + 9 y = 2 cos (3 t)

x \to 0 lim (- \frac{2}{x})

\frac{d}{d x} (\frac{x}{x ^{2} + y ^{2} + z ^{2}})

\int_{0}^{1} (e^{x} + 1)^{2} d x

\int 3 - x x^{2} d x

What is the integral of cotan(x/3) ?
The integral of cotan(x/3) is -3coln|cos(x/3)|+C