What is the integral of csc(2y) ?

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

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integral of csc(2y)

\int csc (2 y) d y

\frac{1}{2} ln ∣ tan (y) ∣ + C

Solution steps

\int csc (2 y) d y

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 y}{2})

Simplify

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

= \frac{1}{2} ln ∣ tan (y) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (y) ∣ + C

\frac{\partial}{\partial x} (y^{2} - x^{2})

ln (cos (x))

d y = (x - 2 y (cot (2 x))) d x

\frac{d}{d x} (lo g_{10} (1 + x))

\frac{d}{d x} (tan (x) + 3 co tan (x))