What is the integral of csc(6x) ?

The integral of csc(6x) is 1/6 ln|tan(3x)|+C

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integral of csc(6x)

\int csc (6 x) d x

\frac{1}{6} ln ∣ tan (3 x) ∣ + C

Solution steps

\int csc (6 x) d x

Apply u-substitution

: \int csc (u) \frac{1}{6} d u

= \int csc (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int csc (u) d u

Apply u-substitution

: \int \frac{1}{v} d v

= \frac{1}{6} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{6} ln ∣ v ∣

Substitute back

= \frac{1}{6} ln tan (\frac{6 x}{2})

Simplify

\frac{1}{6} ln tan (\frac{6 x}{2}) : \frac{1}{6} ln ∣ tan (3 x) ∣

= \frac{1}{6} ln ∣ tan (3 x) ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ tan (3 x) ∣ + C

\frac{2 s}{s ^{2} + 2 s + 1}

\int_{0}^{\frac{π}{4}} (sin (x)) d x

\frac{d}{d x} (\frac{x}{x + 1})

f (x) = 8 x (x + 2), at x = 3