What is the integral of csc(-4x) ?

The integral of csc(-4x) is -1/4 ln|tan(2x)|+C

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integral of csc(-4x)

\int csc (- 4 x) d x

- \frac{1}{4} ln ∣ tan (2 x) ∣ + C

Solution steps

\int csc (- 4 x) d x

Apply u-substitution

: \int - \frac{1}{4} csc (u) d u

= \int - \frac{1}{4} csc (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 csc (u) d u

Apply u-substitution

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

= - \frac{1}{4} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{4} ln ∣ v ∣

Substitute back

= - \frac{1}{4} ln tan (\frac{- 4 x}{2})

Simplify

- \frac{1}{4} ln tan (\frac{- 4 x}{2}) : - \frac{1}{4} ln ∣ tan (2 x) ∣

= - \frac{1}{4} ln ∣ tan (2 x) ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ tan (2 x) ∣ + C

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What is the integral of csc(-4x) ?
The integral of csc(-4x) is -1/4 ln|tan(2x)|+C