What is the integral of csc(8x) ?

The integral of csc(8x) is 1/8 ln|tan(4x)|+C

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integral of csc(8x)

\int csc (8 x) d x

\frac{1}{8} ln ∣ tan (4 x) ∣ + C

Solution steps

\int csc (8 x) d x

Apply u-substitution

= \int csc (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int csc (u) d u

Apply u-substitution

= \frac{1}{8} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{8} ln ∣ v ∣

Substitute back

= \frac{1}{8} ln tan (\frac{8 x}{2})

Simplify

\frac{1}{8} ln tan (\frac{8 x}{2}) : \frac{1}{8} ln ∣ tan (4 x) ∣

= \frac{1}{8} ln ∣ tan (4 x) ∣

Add a constant to the solution

= \frac{1}{8} ln ∣ tan (4 x) ∣ + C