What is the integral of (csc(2x)) ?

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

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integral of (csc(2x))

\int (csc (2 x)) d x

\frac{1}{2} ln ∣ tan (x) ∣ + C

Solution steps

\int csc (2 x) d x

Apply u-substitution

: \int csc (u) \frac{1}{2} d u

= \int csc (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int csc (u) d u

Apply u-substitution

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

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

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

Simplify

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

= \frac{1}{2} ln ∣ tan (x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (x) ∣ + C

y^{^{'}} - 2 t y = 12 t^{2} e^{t^{2}}, y (0) = 5

x \to 4 lim (\frac{x ^{2} - 4}{x - 2})

\frac{d}{d x} (\frac{3 x + 2}{3 x - 2})

y = x^{4} + 2 x^{2} - x (1.2)

(x^{3} - 25 x)^{12}

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