What is the integral of cot(4x) ?

The integral of cot(4x) is 1/4 ln|sin(4x)|+C

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integral of cot(4x)

\int cot (4 x) d x

\frac{1}{4} ln ∣ sin (4 x) ∣ + C

Solution steps

\int cot (4 x) d x

Rewrite using trig identities

= \int \frac{cos ( 4 x )}{sin ( 4 x )} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = sin (4 x)

= \frac{1}{4} ln ∣ sin (4 x) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ sin (4 x) ∣ + C

\frac{d}{d x} (3 sin (x))

\int (- x)^{n} d x

\int sec^{9} (x) tan^{3} (x) d x

\int_{0}^{\frac{π}{4}} x sec^{2} (x) d x

f (x) = \frac{- 4}{x}