What is the integral of cot(3x) ?

The integral of cot(3x) is 1/3 ln|sin(3x)|+C

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integral of cot(3x)

\int cot (3 x) d x

\frac{1}{3} ln ∣ sin (3 x) ∣ + C

Solution steps

\int cot (3 x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{3 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = sin (3 x)

= \frac{1}{3} ln ∣ sin (3 x) ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ sin (3 x) ∣ + C

ln (x x^{2} - 5)

\frac{7}{1 - x}

\frac{d}{d x} (2 e^{- x} + x e^{- x})

x^{2} + x + 8

f (x) = ln (e^{x})