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

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

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

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

\frac{1}{2} ln (2 sin^{2} (x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{sin ( 2 x )}{1 - cos ( 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 = 1 - cos (2 x)

= \frac{1}{2} ln ∣ 1 - cos (2 x) ∣

Simplify

\frac{1}{2} ln ∣ 1 - cos (2 x) ∣ : \frac{1}{2} ln (2 sin^{2} (x))

= \frac{1}{2} ln (2 sin^{2} (x))

Add a constant to the solution

= \frac{1}{2} ln (2 sin^{2} (x)) + C

\int_{4}^{8} \frac{1}{x} d x

x \to 3 lim (\frac{x - 3}{3 - x})

x^{2} - 9, 9 - x^{2}

\int t^{3} e^{- t^{4}} d t

x \to 0 lim (x^{2} cos (\frac{5}{x}))

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