What is the integral of 1/(csc(2x))csc(2x)cot(2x) ?

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

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integral of 1/(csc(2x))csc(2x)cot(2x)

\int \frac{1}{csc ( 2 x )} csc (2 x) cot (2 x) d x

\frac{1}{2} ln ∣ sin (2 x) ∣ + C

Solution steps

\int \frac{1}{csc ( 2 x )} csc (2 x) cot (2 x) d x

Simplify

\frac{1}{csc ( 2 x )} csc (2 x) cot (2 x) : cot (2 x)

= \int cot (2 x) d x

Rewrite using trig identities

= \int \frac{cos ( 2 x )}{sin ( 2 x )} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} 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 \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = sin (2 x)

= \frac{1}{2} ln ∣ sin (2 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ sin (2 x) ∣ + C

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\int - 9 x^{5} e^{x^{6}} d x

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\int (20 x + 3) d x

\frac{d}{d x} ((arctan (3 x))^{2})

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